Construction of complex solutions to nonlinear partial differential equations using simpler solutions
aa r X i v : . [ n li n . S I] F e b Construction of complex solutions to nonlinear partialdifferential equations using simpler solutions ⋆ Alexander V. Aksenov , Andrei D. Polyanin Lomonosov Moscow State University, 1 Leninskiye Gory, Main Building,119234 Moscow, Russia Ishlinsky Institute for Problems in Mechanics RAS, 101 VernadskyAvenue, bldg 1, 119526 Moscow, Russia
Abstract.
The paper describes a number of simple but quite effective meth-ods for constructing exact solutions of nonlinear partial differential equations,that involve a relatively small amount of intermediate calculations. The methodsemploy two main ideas: (i) simple exact solutions can serve to construct morecomplex solutions of the equations under consideration and (ii) exact solutions ofsome equations can serve to construct solutions of other, more complex equations.In particular, we propose a method for constructing complex solutions from sim-ple solutions using translation and scaling. We show that in some cases, rathercomplex solutions can be obtained by adding one or more terms to simpler solu-tions. There are situations where nonlinear superposition allows us to construct acomplex composite solution using similar simple solutions. We also propose a fewmethods for constructing complex exact solutions to linear and nonlinear PDEs byintroducing complex-valued parameters into simpler solutions. The effectivenessof the methods is illustrated by a large number of specific examples (over 30 intotal). These include nonlinear heat equations, reaction–diffusion equations, wavetype equations, Klein–Gordon type equations, equations of motion through porousmedia, hydrodynamic boundary layer equations, equations of motion of a liquidfilm, equations of gas dynamics, Navier–Stokes equations, and some other PDEs.Apart from exact solutions to ‘ordinary’ partial differential equations, we also de-scribe some exact solutions to more complex nonlinear delay PDEs. Along withthe unknown function at the current time, u = u ( x, t ) , these equations contain ∗ This is a preprint of the article A.V. Aksenov, A.D. Polyanin, Methods for constructing complex solutions ofnonlinear PDEs using simpler solutions,
Mathematics , 2021, Vol. 9, No. 4, 345; doi: 10.3390/math9040345. he same function at a past time, w = u ( x, t − τ ) , where τ > is the delay time.Furthermore, we look at nonlinear partial functional-differential equations of thepantograph type, which in addition to the unknown u = u ( x, t ) , also contain thesame functions with dilated or contracted arguments, w = u ( px, qt ) , where p and q are scaling parameters. We propose an efficient approach to construct exactsolutions to such functional-differential equations. Some new exact solutions ofnonlinear pantograph-type PDEs are presented. The methods and examples inthe paper are presented according to the principle “from simple to complex”.Keywords: exact solutions, nonlinear PDEs, reaction-diffusion equations,wave type equations, hydrodynamics equations, PDEs with constant and variabledelay, pantograph-type PDEs, functional-differential equations. Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1. Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 31.2. Concept of ‘Exact Solution’ for Nonlinear PDEs . . . . . . . . . . . 4
2. Construction of Complex Solutions from SimpleSolutions by Shift and Scale Transformations . . . . . . . . . . . 52.1. Some Definitions. Simplest Transformations . . . . . . . . . . . . . 52.2. Construction of Complex Solutions from Simpler Solutions.Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3. Generalization to Nonlinear Multidimensional Equations . . . . . . 142.4. Generalization to Nonlinear Systems of Coupled Equations . . . . . 15
3. Construction of Complex Solutions by Adding Terms orCombining Two Solutions . . . . . . . . . . . . . . . . . . . . . . . 163.1. Construction of Complex Solutions by Adding Termsto Simpler Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2. Construction of Compound Solutions (Nonlinear Superpositionof Solutions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 . The Use of Complex-Valued Parameters for ConstructingExact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.1. Linear Partial Differential Equations . . . . . . . . . . . . . . . . . 234.2. Nonlinear Partial Differential Equations . . . . . . . . . . . . . . . 27
5. Using Solutions of Simpler Equations for ConstructSolutions to Complex Equations . . . . . . . . . . . . . . . . . . . 295.1. Nonlinear Partial Differential Equations . . . . . . . . . . . . . . . 295.2. Partial Differential Equations with Delay . . . . . . . . . . . . . . 325.3. Pantograph-Type Partial Differential Equations . . . . . . . . . . . 355.4. Approach for Constructing Exact Solutions of FunctionalPartial Differential Equations . . . . . . . . . . . . . . . . . . . . . 38
6. Brief Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1. Introduction
Exact solutions of nonlinear partial differential equations and methods for theirconstruction are necessary for the development, analysis and verification of variousmathematical models used in natural and engineering sciences, as well as for testingapproximate analytical and numerical methods. There are several basic methodsfor finding exact solutions and constructing reductions of nonlinear partial dif-ferential equations: the method of group analysis of differential equations (themethod of searching for classical symmetries) [1–6], methods for finding for non-classical symmetries [7–10], the direct Clarkson–Kruskal method [11–14], methodsfor generalized separation of variables [13–15], methods for functional separation ofvariables [14, 16–18], the method of differential constraints [13, 14, 19], the methodof truncated Painlev´e expansions [13,20,21], and use of conservation laws to obtainexact solutions [22–24]. The application of these methods requires considerablespecial training and, as a rule, is accompanied by time-consuming analysis and alarge volume of analytical transformations and intermediate calculations. his paper describes a number of simple, but quite effective, methods forconstructing exact solutions of nonlinear partial differential equations, which donot require much special training and lead to a relatively small amount of inter-mediate calculations. These methods are based on the following two simple, butvery important, ideas: • simple exact solutions can serve as a basis for constructing more complexsolutions of the equations under consideration, • exact solutions to some equations can serve as the basis for constructingsolutions to other more complex equations.The effectiveness of the proposed methods is illustrated by a large numberof specific examples. Nonlinear heat equations, reaction-diffusion equations, wavetype equations, Klein–Gordon type equations, equations of motion in porous me-dia, hydrodynamic boundary layer equations, equations of motion of a liquid film,equations of gas dynamics, Navier–Stokes equations and some other PDEs are con-sidered. In addition to exact solutions of ‘ordinary’ partial differential equations,some exact solutions of more complex nonlinear delay PDEs with constant andvariable delay and pantograph-type functional-differential equations with partialderivatives are also described.The methods and examples in the article are presented according to theprinciple “from simple to complex”. For the convenience of a wide audience withdifferent mathematical backgrounds, the authors tried to do their best, whereverpossible, to avoid special terminology. In this article, the term ‘exact solution’ for nonlinear partial differential equationswill be used in cases where the solution is expressed:(i) in terms of elementary functions, functions included in the equation (thisis necessary when the equation contains arbitrary functions), and indefinite or/anddefinite integrals;(ii) through solutions of ordinary differential equations or systems of suchequations.Combinations of cases (i) and (ii) are also allowed. In case (i) the exact olution can be presented in explicit, implicit, or parametric form. Remark 1.
Exact solutions of nonlinear diffusion and wave type PDEscan be found, for example, in [4, 5, 9, 10, 13–18, 25–44].
2. Construction of Complex Solutions from SimpleSolutions by Shift and Scale Transformations
We say that a partial differential equation, F ( x, t, u, u x , u t , u xx , u xt , u tt , . . . ) = 0 , (1)is invariant with respect to a one-parameter invertible transformation, x = X (¯ x, ¯ t, ¯ u, a ) , t = T (¯ x, ¯ t, ¯ u, a ) , u = U (¯ x, ¯ t, ¯ u, a ) , (2)if, after substituting expressions (2) in (1), we obtain exactly the same equation F (¯ x, ¯ t, ¯ u, ¯ u ¯ x , ¯ u ¯ t , ¯ u ¯ x ¯ x , ¯ u ¯ x ¯ t , ¯ u ¯ t ¯ t , . . . ) = 0 . It is important to note that the free parameter a , which can take values ina certain interval ( a , a ) , is not included in the equation (1).Transformations that preserve the form of equation (1) transform a solutionof the considered equation into a solution of the same equation.A function I ( x, t, u ) (different from a constant and independent of a ) is calledan invariant of transformation (2) if it is preserved under this transformation, i.e. I ( x, t, u ) = I (¯ x, ¯ t, ¯ u ) for all admissible values of the parameter a .A solution u = Φ( x, t ) of equation (1) is called invariant if, under thetransformation (2), it transforms into exactly the same solution ¯ u = Φ(¯ x, ¯ t ) .Further, we will consider only one-parameter transformations of the form x = ¯ x + b , t = ¯ t + b , u = ¯ u + b ( translation ); x = c ¯ x, t = c ¯ t, u = c ¯ u ( scaling ) , nd the composition of these transformations. Here b n and c n ( n = 1 , , ) areconstants depending on the free parameter a . Such transformations will be calledthe simplest transformations .The following example shows how to determine the invariants of the simplesttransformations and the form of the corresponding invariant solutions. Example 1.
Consider a transformation consisting of the translation in x and the scaling in t and u : x = ¯ x − m ln a, t = a ¯ t, u = a k ¯ u, (3)where k and m are some constants. Excluding the parameter a , we find twofunctionally independent invariants: I = x + m ln t, I = ut − k . (4)If the considered equation is invariant under transformation (3), then itadmits an invariant solution, which can be represented as I = ϕ ( I ) [1, 4] or u = t k ϕ ( z ) , where z = x + m ln t . Substituting the resulting expression into theoriginal equation we arrive at an ordinary differential equation for the function ϕ = ϕ ( z ) . Simple one-term solutions in the form of a product of functions of different vari-ables are most easily found by the method of separation of variables (the simplestsolutions of this type u = Ax α t β are easily determined from the equations underconsideration by the method of undefined coefficients). The methods for con-structing more complex solutions based on such solutions are described below.First we will consider a simple multiplicative separable solution of the specialform u = t k ϕ ( x ) , (5)where k is some constant and ϕ ( x ) is some function. Such solutions do not change(are invariant) under the scaling transformation t = a ¯ t, u = a k ¯ u. (6) elow, in the form of a proposition, we describe a method that allows us toconstruct more complex solutions based on one-term solutions of the form (5). Proposition 1.
Let the equation F ( t, u, u x , u t , u xx , u xt , u tt , . . . ) = 0 , (7) which does not explicitly depend on the x , has a simple solution of the form (5) and does not change under the scaling transformation (6) (i.e., equation (7) hasthe same property as the original solution (5) ). Then this equation also has amore complex solution of the form u = t k ϕ ( z ) , z = x + m ln t, (8) where m is an arbitrary constant. Proof.
Equation (7) does not explicitly depend on the spatial variable andis invariant under the translation in x . Consider transformation (3), which is acomposition of the translation in x and scaling in t and u (see (6)). Transforma-tion (3) has two functionally independent invariants (4). Therefore, the solutioninvariant under transformation (3) has the form (8) (see Example 1). Remark 2.
In general, the form of functions ϕ ( x ) and ϕ ( x ) , which are in-cluded in the original solution (5) and the more complex solution (8), respectively,may differ and ψ ( z ) | m =0 = ϕ ( x ) .Consider now a simple multiplicative separable solution of the special form u = x n ψ ( t ) , (9)where n is some constant and ψ ( t ) is some function. Such solutions do not change(are invariant) under the scaling transformation x = a ¯ x, u = a n ¯ u. (10)A more complex solution than (9) can be obtained by using Proposition 1,redefining the constants and variables in (5)–(8) accordingly. A different, butequivalent method for constructing a more complex solution is described below,which is sometimes more convenient to use in practice. Proposition 2.
Let the equation F ( x, u, u x , u t , u xx , u xt , u tt , . . . ) = 0 , (11) hich does not explicitly depend on t , has a simple solution of the form (9) , anddoes not change under the scaling transformation (10) (i.e., equation (11) has thesame property as the original solution (9) ). Then this equation also has a morecomplex solution of the form u = e − npt ψ ( y ) , y = xe pt , (12) where p is an arbitrary constant. Proof.
Equation (11) is invariant under the translation in t . Consider atransformation that is a composition of the translation in t and the scaling in x and u (see (10)): x = a ¯ x, t = ¯ t − p ln a, u = a n ¯ u, (13)where p is an arbitrary constant ( p = 0 ). Transformation (13) has two functionallyindependent invariants I = y = xe pt and I = e npt u . Therefore, a solutioninvariant with respect to transformation (13) has the form (12).Obviously, equation (11) admits the degenerate solution (12) with p = 0 . Remark 3.
In (12), the function argument y is linear in x . Therefore,solution (12) is easy to differentiate with respect to x . This solution representationshould be used for equations which contain partial derivatives with respect to x of a higher order than with respect to t . Example 2.
Consider the Boussinesq equation u t = a ( uu x ) x , (14)which describes the unsteady flow of groundwater in a porous medium with a freesurface [46].Equation (14) has a simple exact solution, u = − x at , (15)which is simultaneously a solution of two types (5) and (9). Let us consider in orderboth possibilities of constructing more complex solutions based on solution (15). ◦ . Solution (15) and equation (14) retain their form under the scaling trans-formation t = c ¯ t , u = ¯ u/c . Therefore, by virtue of Proposition 1 equation (14) dmits a more complex exact solution, u = ϕ ( z ) t , z = x + k ln t, where the function ϕ = ϕ ( z ) satisfies the ordinary differential equation (here-inafter ODE): kϕ ′ z − ϕ = a ( ϕϕ ′ z ) ′ z . (16)Note that equation (16) for k = 0 admits a one-parameter family of solutionsin the form of a quadratic polynomial, ϕ = − x a + C x − aC , where C is an arbitrary constant. For C = 0 this solution coincides with theoriginal solution (15). ◦ . Solution (15) and equation (14) retain their form also under the scalingtransformation x = c ¯ x , u = c ¯ u . Therefore, by virtue of Proposition 2, equa-tion (14) admits another exact solution u = e − pt ψ ( y ) , y = xe pt , where p is an arbitrary constant and the function ψ = ψ ( y ) is described by theODE: pyψ ′ y − pψ = a ( ψψ ′ y ) ′ y . Example 3.
Consider now the Guderley equation u xx = au y u yy , (17)which is used to describe transonic gas flows [47].Equation (17) admits a simple exact solution, u = y ax , (18)which is a special case of two types of solutions (5) and (9). Let us consider in orderboth possibilities of constructing more complex solutions based on solution (18). ◦ . Solution (18) and equation (17) retain their form under the scaling trans-formation x = c ¯ x , u = c − ¯ u . Therefore, by virtue of Proposition 1, equation (17)has a more complex exact solution of the form u = x − ϕ ( z ) , z = y + m ln x, here the function ϕ = ϕ ( z ) is described by the second-order ODE: m ϕ ′′ zz − mϕ ′ z + 6 ϕ = aϕ ′ z ϕ ′′ zz . For m = 0 , this equation admits a one-parameter family of solutions in the formof a cubic polynomial, ϕ ( z ) = z a + C z + aC z + a C , where C is an arbitrary constant. For C = 0 this solution coincides with theoriginal solution (18). ◦ . Solution (18) and equation (17) retain their form also under the scalingtransformation y = c ¯ y , u = c ¯ u . Therefore, by virtue of Proposition 2, one canalso obtain another more complex exact solution, u = e − px ψ ( z ) , z = ye px , where p is an arbitrary constant and the function ψ = ψ ( z ) is described by theODE: p z ψ ′′ zz − p zψ ′ z + 9 p ψ = aψ ′ z ψ ′′ zz . Example 4.
In gas dynamics, there is a nonlinear wave equation, u tt = a ( u b u x ) x , b = 0 , (19)which admits a simple exact solution of the form u = a − /b x /b t − b . (20)This solution belongs to both classes of solutions (5) and (9). Therefore, based onsolution (20), we can construct two more complex solutions described below. ◦ . Solution (20) and equation (19) are invariant under the scaling transfor-mation t = c ¯ t , u = c − /b ¯ u . By virtue of Proposition 1, equation (19) has a morecomplex solution of the form u = t − /b ϕ ( z ) , z = x + m ln t, where the function ϕ = ϕ ( z ) is described by the ODE: m ϕ ′′ zz − m ( b + 4) b ϕ ′ z + 2( b + 2) b ϕ = a ( ϕ b ϕ ′ z ) ′ z . ◦ . Solution (20) and equation (19) are also invariant under the scaling trans-formation x = c ¯ x , u = c /b ¯ u . Therefore, by virtue of Proposition 2, equation (19)admits another solution u = e − pt/b ψ ( y ) , y = xe pt , where p is an arbitrary constant and the function ψ = ψ ( y ) satisfies the second-order nonlinear ODE: p y ψ ′′ yy + p ( b − b yψ ′ y + 4 p b ψ = a ( ψ b ψ ′ y ) ′ y . Example 5.
The system of boundary layer equations on a flat plate byintroducing a stream function is reduced to one nonlinear third-order PDE: u y u xy − u x u yy = νu yyy , (21)where ν is the kinematic viscosity of the fluid [45].Equation (21) has a simple solution of the form u = 6 νxy , (22)which generates two more complex solutions. ◦ . Solution (22) and equation (21) do not change under the scaling trans-formation x = a ¯ x , u = a ¯ u . Therefore, by virtue of Proposition 1, equation (21)has a more complex solution of the form u = xϕ ( z ) , z = y + k ln x, where k is an arbitrary constant and the function ϕ = ϕ ( z ) satisfies the ODE: − ϕϕ ′′ zz + ( ϕ ′ z ) = νϕ ′′′ zzz . ◦ . Solution (22) and equation (21) do not change also under the scalingtransformation y = a ¯ y , u = ¯ u/a . Therefore, by virtue of Proposition 2, equa-tion (21) admits another solution u = e px ψ ( z ) , z = ye px , where p is an arbitrary constant and the function ψ = ψ ( z ) is described by theODE: pψψ ′′ zz − p ( ψ ′ z ) = νψ ′′′ zzz . xample 6. Consider a fourth-order nonlinear evolution equation describ-ing the change in the film thickness of a heavy viscous liquid moving along ahorizontal superhydrophobic surface with a variable surface tension coefficient, u t = [( au + bx / u )( u x − c ( x u xx ) x )] x , (23)where a , b , and c are some constants [48, 49].Equation (23) has a simple solution of the form u = x / f ( t ) , (24)where the function f = f ( t ) is described by the first-order ODE with separablevariables f ′ t = (2 c + 9) f ( af + b ) . Solution (24) and equation (23) are invariant under the scaling transforma-tion x = k ¯ x , u = k / ¯ u . Therefore, by virtue of Proposition 2, equation (23) alsohas a more complex solution of the form u = e − pt/ ψ ( y ) , y = xe pt , where the function ψ = ψ ( y ) satisfies the fourth-order nonlinear ODE [48, 49]: pyψ ′ y − pψ = [( aψ + by / ψ )( ψ ′ y − ( cy ψ ′′ yy ) ′ y ] ′ y , where p is an arbitrary constant. Example 7.
Consider a n th-order nonlinear PDE of the form u t = u s F ( u x /u, u xx /u, . . . , u ( n ) x /u ) . s = 1 . (25)Equation (25) has a simple solution, u = t / (1 − s ) ϕ ( x ) , (26)where the function ϕ = ϕ ( x ) is described by the ODE: ϕ − s = ϕ s F ( ϕ ′ x /ϕ, ϕ ′′ xx /ϕ, . . . , ϕ ( n ) x /ϕ ) . Solution (26) and equation (25) are invariant under the scaling transforma-tion t = a ¯ t , u = a / (1 − s ) ¯ u . Therefore, by virtue of Proposition 1, equation (25)also has a more complex solution of the form u = t / (1 − s ) ϕ ( z ) , z = x + m ln t, here m is an arbitrary constant and the function θ = θ ( z ) satisfies the ODE: mϕ ′ z + ϕ − s = ϕ s F ( ϕ ′ z /ϕ, ϕ ′′ zz /ϕ, . . . , ϕ ( n ) z /ϕ ) . Proposition 3.
Let the equation F ( u, u x , u t , u xx , u xt , u tt , . . . ) = 0 , (27) which does not explicitly depend on x and t (and therefore admits the traveling-wave solution [13]) does not change under scaling of the unknown function u = c ¯ u, (28) where c > is an arbitrary constant. Then this equation admits an exact solution(more complicated than the traveling-wave solution) of the form u = e kt ϕ ( z ) , z = px + qt, (29) where k , p , and q are arbitrary constants ( pq = 0 ). Proof.
Consider a transformation that is a composition of translations in x and t and scaling of the unknown function (28): x = ¯ x + 1 p ln a, t = ¯ t − q ln a, u = a − k/q ¯ u, (30)where a > is an arbitrary constant ( c = a − k/q ), p and q are some constants( pq = 0 ). The transformation (30) preserves the form of equation (27) and has twofunctionally independent invariants I = z = px + qt and I = e − kt u . Therefore, asolution that is invariant with respect to transformation (30), can be representedas (29). Solution of the form (29) is obtained from the invariant solution byapplying the scaling transformation (28). Example 8.
Consider the nonlinear heat-type equation u t = au xx + uf ( u x /u ) , (31)where f = f ( ξ ) is an arbitrary function.Equation (31) is invariant under the scaling transformation (28). Therefore,by virtue of Proposition 3, this equation has a solution of the form (29), wherethe function ϕ = ϕ ( z ) satisfies the nonlinear ODE: kϕ + qϕ ′ z = ap ϕ ′′ zz + ϕf ( pϕ ′ z /ϕ ) . xample 9. Consider a more complex nonlinear PDE of order n , u t = uF ( u x /u, u xx /u, . . . , u ( n ) x /u ) . (32)Equation (32) is invariant under the scaling transformation (28). Therefore,by virtue of Proposition 3, this equation has a solution of the form (29), wherethe function ϕ = ϕ ( z ) satisfies the nonlinear ODE: kϕ + qϕ ′ z = ϕF ( pϕ ′ z /ϕ, p ϕ ′′ zz /ϕ, . . . , p n ϕ ( n ) z /ϕ ) , where F ( w , w , . . . , w n ) is an arbitrary function. The above Propositions 1–3 allow obvious generalizations to the case of an arbi-trary number of spatial variables.
Example 10.
Consider the nonlinear heat equation with n spatial variables u t = a n X i =1 ∂∂x i (cid:18) u k ∂u∂x i (cid:19) , k = 0 . (33)Equation (33) admits a simple multiplicative separable solution, u = t − /k ϕ ( x , . . . , x n ) , (34)where the function ϕ = ϕ ( x , . . . , x n ) satisfies the stationary equation − k ϕ = a n X i =1 ∂∂x i (cid:18) ϕ k ∂ϕ∂x i (cid:19) . Solution (34) and equation (33) are invariant under the scaling transforma-tion t = c ¯ t , u = c − k ¯ u . Therefore, by virtue of Proposition 1, equation (33) alsohas a more complex solution of the form u = t − /k θ ( z , . . . z n ) , z i = x i + m i ln t, where m i are arbitrary constants, and the function θ = θ ( z , . . . z n ) satisfies thestationary equation − k θ + n X i =1 m i ∂θ∂z i = a n X i =1 ∂∂z i (cid:18) θ k ∂θ∂z i (cid:19) . .4. Generalization to Nonlinear Systems of Coupled Equations The above Propositions 1–3 can also be used to find exact solutions of systems ofcoupled PDEs.
Example 11.
Consider the nonlinear system consisting of two coupledreaction-diffusion equations u t = a ( u b u x ) x + uf ( u/v ) ,v t = a ( v b v x ) x + vg ( u/v ) , (35)where a and b are some constants ( b = 0 ), and f ( z ) and g ( z ) are arbitraryfunctions.System of equations (35) has a simple solution of the form u = x /b ϕ ( t ) , v = x /b ψ ( t ) , (36)where the functions ϕ = ϕ ( t ) and ψ = ψ ( t ) are described by the system of first-order ODEs: ϕ ′ t = 2 a ( b + 2) b ϕ b +1 + ϕf ( ϕ/ψ ) ,ψ ′ t = 2 a ( b + 2) b ψ µ +1 + ψg ( ϕ/ψ ) . Solution (36) and system of equations (35) are invariant under the scalingtransformation x = c ¯ x , u = c /b ¯ u , v = c /b ¯ v . Therefore, by virtue of Proposition 2,the system of equations (35) also has a more complex solution of the form u = e − mt/b Φ( z ) , v = e − mt/b Φ( z ) , z = xe mt , where the functions Φ = Φ( z ) and Ψ = Ψ( z ) are described by the ODE system: mz Φ ′ z − mb Φ = a (Φ b Φ ′ z ) ′ z + Φ f (Φ / Ψ) ,mz Ψ ′ z − mb Ψ = a (Ψ b Ψ ′ z ) ′ z + Ψ f (Φ / Ψ) , where m is an arbitrary constant. Example 12.
Consider another nonlinear system consisting of two coupledreaction-diffusion equations u t = a ( u b u x ) x + u b +1 f ( u/v ) ,v t = a ( v b v x ) x + v b +1 g ( u/v ) , (37) here a and b are some constants ( b = 0 ), and f ( z ) and g ( z ) are arbitraryfunctions.System of equations (37) has a simple solution of the form u = t − /b ϕ ( x ) , v = t − /b ψ ( x ) , (38)where the functions ϕ = ϕ ( x ) and ψ = ψ ( x ) are described by the second-orderODE system − ϕb = a ( ϕ b ϕ ′ x ) ′ x + ϕ b +1 f ( ϕ/ψ ) , − ψb = a ( ψ b ψ ′ x ) ′ x + ψ b +1 g ( ϕ/ψ ) . Solution (38) and system of equations (37) are invariant under the scalingtransformation t = c ¯ t , u = c − /b ¯ u , v = c − /b ¯ v . By virtue of Proposition 1, thesystem of equations (37) also has a more complex solution of the form u = t − /b Φ( z ) , v = t − /b Ψ( z ) , z = x + m ln t, where m is an arbitrary constant, and the functions Φ = Φ( z ) and Ψ = Ψ( z ) satisfy the system ODE: − Φ b + m Φ ′ z = a (Φ b Φ ′ z ) ′ z + Φ b +1 f (Φ / Ψ) , − Ψ b + m Ψ ′ z = a (Ψ b Ψ ′ z ) ′ z + Ψ b +1 g (Φ / Ψ) .
3. Construction of Complex Solutions by Adding Terms orCombining Two Solutions
In some cases, simple solutions can be generalized by adding one or more ad-ditional terms to them, which leads to more complex solutions with generalizedseparation of variables [13–15]. We demonstrate the possible course of reasoningin such cases using the examples of the Boussinesq equation (14) and the Guderleyequation (17). xample 13. As mentioned earlier, the Boussinesq equation (14) has asolution with a simple separation of variables (quadratic in x , see (15)), which wewrite as u = ϕ ( t ) x , ϕ ( t ) = − / (6 at ) . (39)Let’s try to find a more complex solution in the form of the sum u ( x, t ) = ϕ ( t ) x + ψ ( t ) x k , k = 2 , (40)whose first term coincides with the solution (39). The second term of formula (40)includes the function ψ ( t ) and the coefficient k , which must be found.Substituting (40) in (14), after elementary transformations we get ( ϕ ′ t − aϕ ) x + [ ψ ′ t − a ( k + 1)( k + 2) ϕψ ] x k − ak (2 k − ψ x k − = 0 . (41)Since this equality must hold identically for any x , the functional coefficientsfor various powers of x in (41) must be zero. Thus, there are two possible cases k = 0 and k = 1 / (both correspond to the vanishing of the coefficient at x k − ),which must be considered separately. ◦ . The first case . Substituting k = 0 into (41), to define the functions ϕ = ϕ ( t ) and ψ = ψ ( t ) , we have the system of ODEs: ϕ ′ t − aϕ = 0 , ψ ′ t − aϕψ = 0 , the general solution of which is determined by the formulas ϕ ( t ) = − a ( t + C ) , ψ ( t ) = C | t + C | / , (42)where C and C are arbitrary constants. ◦ . The second case (the Barenblatt–Zeldovich dipole solution [50]). Sub-stituting k = 1 / into (41), we obtain a system of ODEs for determining thefunctions ϕ = ϕ ( t ) and ψ = ψ ( t ) : ϕ ′ t − aϕ = 0 , ψ ′ t − aϕψ = 0 . The general solution of this system is ϕ ( t ) = − a ( t + C ) , ψ ( t ) = C | t + C | / . (43) iven the formulas (40), (42), (43), as a result, we obtain two three-parameter generalized separable solutions of equation (14): u = − a ( t + C ) ( x + C ) + C | t + C | / ,u = − a ( t + C ) ( x + C ) + C | t + C | / ( x + C ) / , where for the sake of greater generality, an arbitrary translation in x is additionallyadded. Remark 4.
The wave type equation with quadratic nonlinearity u tt = a ( uu x ) x , also admits solutions of the form (40) with k = 0 and k = 1 / . Example 14.
Let us now return to the Goderley equation (17). Thisequation admits the simple exact solution (18), which we write in the form u = f ( x ) y , f ( x ) = 1 / (3 ax ) . We will look for more complex solutions (with generalized separation ofvariables) equation (17) in the form u ( x, y ) = ϕ ( x ) y k + ψ ( x ) , (44)where the functions ϕ ( x ) and ψ ( x ) and the constant k = 0 are determined in thesubsequent analysis (solution (18) is a particular case of solution (44) for k = 3 and ψ = 0 ).It is important to note that binomial solutions of the form (44) are quiteoften encountered in practice and are the simplest generalized separable solutionsof nonlinear PDEs.Substituting (44) in (17), after rearranging the terms, we come to the relation ϕ ′′ xx y k − ak ( k − ϕ y k − + ψ ′′ xx = 0 , (45)which contains the power functions y k and y k − and must be satisfied identicallyfor any y .Consider two cases: ψ ′′ xx = 0 and ψ ′′ xx = 0 . ◦ . The first case . When ψ ′′ xx = 0 we get a binomial equation with separablevariables, which can be satisfied if we set k = 3 , ϕ ′′ xx − aϕ = 0 . (46)The general solution of the autonomous ODE (46) can be represented in the im-plicit form x = ± Z (12 aϕ + C ) − / dϕ + C . Moreover, this equation admits a particular solution of the power form ϕ = a ( x + C ) − , which leads to a three-parameter exact solution of equation (17): u = 13 a ( x + C ) − y + C x + C , (47)where C , C , and C are arbitrary constants. ◦ . Second case . To balance the function ψ ′′ xx = 0 with second term inequality (45), we must set k = 3 / . As a result, we obtain a binomial equation,which can be satisfied by setting ϕ ′′ xx = 0 , ψ ′′ xx = aϕ . These equations are easily integrated and lead to a four-parameter exact solutionof equation (17): u = ( C x + C ) y / + 3 a C ( C x + C ) + C x + C , (48)where C , C , C , and C are arbitrary constants. Example 15.
Let us return to the hydrodynamic boundary layer equa-tion (21). It is easy to verify that this equation admits the self-similar solution [53]: u = F ( ξ ) , ξ = y/x, (49)where the function F = F ( ξ ) satisfies the third-order ODE: − ( F ′ z ) = νF ′′′ zzz .We look for a more general solution of equation (21) by adding the function ϕ ( x ) to (49): u = F ( ξ ) + ϕ ( x ) , ξ = y/x. imple calculations show that ϕ ( x ) = a ln x , where a is an arbitrary constant. Asa result, we obtain a non-self-similar solution of the boundary layer equation (21)of the form [13]: u = F ( ξ ) + a ln x, ξ = y/x, where the function F = F ( ξ ) is described by the third-order ODE: − ( F ′ z ) − aF ′′ zz = νF ′′′ zzz . In some cases, two similar but different solutions of the considered nonlinear PDEcan be combined to obtain a more general composite solution. We demonstrate thepossible course of reasoning in such cases by examples of the Goderley equationand the nonlinear diffusion equation with a second-order volume reaction.
Example 16.
From expressions (47) and (48) it follows that the Guderleyequation (17) has two solutions of the same type u = ϕy / + ψ and u = ϕy + ψ ,which differ from each other by the exponent y . This circumstance suggests anattempt to construct a more general solution of equation (17), that includes bothterms with different exponents at once. In other words, we are looking for acomposite solution of the form u ( x, y ) = ϕ ( x ) y + ϕ ( x ) y / + ψ ( x ) . (50)Substitute it in equation (17). After combining the functional factors for power-functions y n/ ( n = 0 , , ), we get ( ϕ ′′ − aϕ ) y + ( ϕ ′′ − aϕ ϕ ) y / + ψ ′′ − aϕ = 0 . For this equality to hold for any y , it is necessary to equate the functional factorsof y n/ to zero. As a result, we arrive at the system of ODEs: ϕ ′′ − aϕ = 0 ,ϕ ′′ − aϕ ϕ = 0 ,ψ ′′ − aϕ = 0 . (51)Thus, it is constructively proved that equation (17) admits a solution of theform (50) (this solution was obtained in [51]). t can be shown that the system (51) admits the exact solution ϕ = 13 a ( x + C ) − ,ϕ = C ( x + C ) / + C ( x + C ) − / ,ψ = 3 a C ( x + C ) + 38 aC C ( x + C ) + 916 aC ( x + C ) − + C x + C . Example 17.
Let us now consider a nonlinear diffusion equation with thesecond-order volume reaction u t = a ( uu x ) − bu . (52)The procedure for constructing a composite solution of this equation will be carriedout in two stages: first, we will find two fairly simple solutions, and then, usingthese solutions, we will construct a composite solution. ◦ . Solutions of exponential form in x . Exact generalized separable solutionsof equation (52) are sought in the form u ( x, t ) = ϕ ( t ) e λx + ψ ( t ) , (53)where functions ϕ = ϕ ( t ) and ψ = ψ ( t ) and the constant λ are to be determinedin the subsequent analysis. Substituting (53) in (52) and collecting similar termsat exponents e nλx ( n = 0 , , ), we get ( b − aλ ) ϕ e λx + [ ϕ ′ t + (2 b − aλ ) ϕψ ] e λx + ψ ′ t + bψ = 0 . Since this equality must be satisfied identically for any x , the functional factorsof e nλx must be equated to zero. As a result, we come to the differential-algebraicsystem b − aλ = 0 ,ϕ ′ t + (2 b − aλ ) ϕψ = 0 ,ψ ′ t + bψ = 0 , which allows two solutions λ = ± (cid:18) b a (cid:19) / , ϕ = C | t + C | / , ψ = 1 b ( t + C ) , (54)where C and C are arbitrary constants. ◦ . Composite solution of exponential form in x . From relations (53)and (54) it follows that equation (52) has two solutions u , = ϕe ± λx + ψ . Theydiffer in structure from each other only by the sign of the exponent λ .This circumstance suggests trying to construct a more general solution ofequation (52), which includes both exponential terms at once. In other words, weare looking for a composite solution of the form u ( x, t ) = ϕ ( t ) e − λx + ϕ ( t ) e λx + ψ ( t ) , λ = (cid:18) b a (cid:19) / . (55)Substituting (55) in (52), after elementary transformations we have [( ϕ ) ′ t + bϕ ψ ] e − λx + [( ϕ ) ′ t + bϕ ψ ] e λx + ψ ′ t + b (2 ϕ ϕ + ψ ) = 0 . Equating the functional factors of e nλx ( n = 0 , ± ) to zero, we arrive at thefirst-order ODE system ( ϕ ) ′ t + bϕ ψ = 0 , ( ϕ ) ′ t + bϕ ψ = 0 ,ψ ′ t + b (2 ϕ ϕ + ψ ) = 0 . (56)Thus, it has been proved that equation (52) admits the solution of the form (55).By excluding ψ from the first two equations in (56), we obtain the equality ( ϕ ) ′ t /ϕ = ( ϕ ) ′ t /ϕ . This implies that ϕ = Aϕ ( t ) , ϕ = Bϕ ( t ) , where A and B are arbitrary constants. Therefore, the generalized separable solution (55) isreduced to the form u ( x, t ) = ϕ ( t )( Ae − λx + Be λx ) + ψ ( t ) , λ = (cid:18) b a (cid:19) / , (57)where the functions ϕ = ϕ ( t ) and ψ = ψ ( t ) are described by the nonlinear systemof two ODEs: ϕ ′ t + bϕψ = 0 ,ψ ′ t + b (2 ABϕ + ψ ) = 0 . (58)By excluding t , this autonomous system is reduced to one ODE, which is homoge-neous and therefore can be integrated [52]. Note that the system of equations (58)for AB > admits two simple solutions ϕ = ± b √ AB ( t + C ) , ψ = 23 b ( t + C ) , hich define the solution (57) in the form of a product of functions of differentarguments. ◦ . Solution of trigonometric type in x . When writing formulas (55) and (57)implicitly it was assumed that ab > . For ab < we have λ = iβ, β = (cid:18) − b a (cid:19) / , i = − . In this case, in solution (57) instead of exponential functions, trigonometric func-tions appear, i.e. it can be represented in the form u ( x, t ) = ϕ ( t )[ A cos( βx ) + B sin( βx )] + ψ ( t ) , β = (cid:18) − b a (cid:19) / , (59)where A and B are arbitrary constants. Substituting (59) into equation (52)and performing calculations similar to those in Item ◦ , we obtain the followingnonlinear system of ODEs for the functions ϕ = ϕ ( t ) and ψ = ψ ( t ) : ϕ ′ t + bϕψ = 0 ,ψ ′ t + b [ ( A + B ) ϕ + ψ ] = 0 . (60)This system allows for two simple solutions ϕ = ± b p A + B ( t + C ) , ψ = 23 b ( t + C ) , which determine the solution (59) in the form of a product of functions of differentarguments.
4. The Use of Complex-Valued Parameters for ConstructingExact Solutions
In the case of linear partial differential equations, the following proposition can beused to construct more complex solutions from simpler solutions.
Proposition 4.
Let a linear homogeneous PDE with two independent vari-ables x and t have a one-parameter solution of the form u = ϕ ( x, t, c ) , where c s a parameter that is not included in the original equation. Then the consideredequation also has two two-parameter solutions u = Re ϕ ( x, t, a + ib ) , u = Im ϕ ( x, t, a + ib ) , (61) where a and b are arbitrary real constants, Re z and Im z are real and imaginaryparts of complex number z . Proof.
The validity of the proposition follows from the linearity of theequation and from the fact that the solution u = ϕ ( x, t, c ) is also a solution for c = a + ib .Proposition 4 implies the validity of the following two consequences: Corollary 1.
Let a linear homogeneous PDE not depend explicitly on theindependent variable t and have a solution u = ϕ ( x, t ) . Then this equation alsohas two one-parameter families of solutions u = Re ϕ ( x, t + ia ) , u = Im ϕ ( x, t + ia ) , where a is an arbitrary real constant. Corollary 2.
Let a linear homogeneous PDE not depend explicitly on theindependent variable x and have a solution u = ϕ ( x, t ) . Then this equation alsohas two one-parameter families of solutions u = Re ϕ ( x + ia, t ) , u = Im ϕ ( x + ia, t ) , where a is an arbitrary real constant. Example 18.
Consider the linear heat equation u t − u xx = 0 . (62)It is easy to verify that this equation admits an exact solution of the exponentialform u = exp( c t + cx ) , where c is an arbitrary parameter.Using Proposition 4, we obtain two more complicated two-parameter familiesexact solutions of equation (62): u = Re exp( c t + cx ) | c = a + ib = exp[( a − b ) t + ax ] cos[ b (2 at + x )] ,u = Im exp( c t + cx ) | c = a + ib = exp[( a − b ) t + ax ] sin[ b (2 at + x )] . xample 19. Consider the linear wave equation u tt − u xx = 0 . (63)It is easy to verify that equation (63) admits translation transformations for bothindependent variables and has the particular solution u = xx − t . (64)Making a translation in solution (64) with an imaginary parameter in t and using Corollary 1, we find two more complicated one-parameter families ofsolutions to equation (63): u = Re xx − ( t + ia ) = x ( x − t + a )( x − t + a ) + 4 a t ,u = Im xx − ( t + ia ) = 2 axt ( x − t + a ) + 4 a t . Making a translation in solution (64) with an imaginary parameter in x andusing Corollary 2, we find two other one-parameter families of solutions: u = Re x + ia ( x + ia ) − t = x ( x − t + a )( x − t − a ) + 4 a x ,u = Im x + ia ( x + ia ) − t = a ( x + t + a )( x − t − a ) + 4 a x . Example 20.
Consider the linear heat equation u t = u xx + 1 x u x , (65)which describes two-dimensional processes with axial symmetry, where x is theradial coordinate. It is easy to verify that equation (65) admits a translationtransformation with respect to the variable t and has the particular solution u = 1 t exp (cid:16) − x t (cid:17) . (66)Making a translation in solution (66) with an imaginary parameter in thevariable t and using Corollary 1, we find two more complicated one-parameter amilies of solutions: u = Re 1 t + ia exp (cid:18) − x t + ia ) (cid:19) == 1 t + a exp (cid:18) − x t t + a ) (cid:19)(cid:18) t cos ax t + a ) + a sin ax t + a ) (cid:19) ,u = Im 1 t + ia exp (cid:18) − x t + ia ) (cid:19) == 1 t + a exp (cid:18) − x t t + a ) (cid:19)(cid:18) a cos ax t + a ) − t sin ax t + a ) (cid:19) . Example 21.
Consider the linear wave equation with variable coefficients u tt − ( xu x ) x = 0 . (67)This equation admits a translation transformation with respect to t and has theexact solution u = C t (4 x − t ) / , (68)where C is an arbitrary constant.By making a translation in solution (68) an imaginary parameter in t andusing the Corollary 1, we can find two more complicated one-parameter familiesof solutions by formulas u = Re C ( t + ia )(4 x − ( t + ia ) ) / , u = Im C ( t + ia )(4 x − ( t + ia ) ) / . (69)The final form of these solutions is not presented here, due to the cumbersomenessof their recording. The solution u was obtained in [55] and was used to describethe propagation of localized disturbances in one-dimensional shallow water overan inclined bottom. Note, that in [56], another exact solution of the equation (67)was obtained by integrating the parameter a . Proposition 5.
Let a linear homogeneous PDE have a one-parameter so-lution of the form u = ϕ ( x, t, c ) , where c is a real parameter that is not includedin the equation. Then, by n -fold differentiation or integration of this solution, onecan obtain other exact solutions of the considered equation [57, 58]. Corollary.
Exact solutions of linear PDEs, which do not explicitly dependon the independent variable t , can be constructed by differentiating or/and inte- rating with respect to parameters a and b in solutions (61) that are obtained byintroducing the complex parameter c = a + ib . Remark 5.
In [56] by integrating formulas (69) with parameter a it wasobtained a new solution of equation (67). ◦ . In some cases, it is possible to obtain another solution from one solution,passing from real parameters to complex ones in such a way that the transformedequation and the solution remain real. Let’s explain this with a few examples. Example 22.
Let us return again to equation (52). It is easy to verify thatits trigonometric solution (59) and the system of equations (60) can be obtainedfrom the solution exponential form (57) and system of equations (58), if in thelatter we formally set e λx = e iβx = cos( βx ) + i sin( βx ) , e − λx = e − iβx = cos( βx ) − i sin( βx ) ,A = ( A + iB ) , B = ( A − iB ) , A = A + B, B = i ( B − A ) . (70) Example 23.
Consider the equation u t = au xx + uf ( u x − bu ) , (71)where f ( w ) is an arbitrary function.It is easy to verify that equation (71) has the simple multiplicative separablesolution (exponential in x ): u = ψ ( t ) e λx , (72)where the parameter λ and the function ψ ( t ) are to be determined in the subse-quent analysis. Substituting (72) in (71), we obtain two solutions of the form (72),where λ = ±√ b, ψ ′ t = [ ab + f (0)] ψ. The presence of two solutions of the same type corresponding to ± λ suggeststrying them ‘combine’ and look for a more general composite solution of the form u = ψ ( t )( Ae λx + Be − λx ) , (73) here A and B are some constants. Substituting (73) in Eq. (71), we obtain asolution of the form (73), where A and B are arbitrary constants, and the function ψ = ψ ( t ) satisfies the nonlinear ODE: ψ ′ t = abψ + ψf ( − ABbψ ) . (74)Substituting (70) into (73) and (74), we arrive at a new solution containingalready trigonometric functions in x , u = ϕ ( t )[ A cos( βx ) + B sin( βx )] , β = √− b, where A and B are arbitrary constants, and the function ψ = ψ ( t ) is describedby a nonlinear ODE: ψ ′ t = abψ + ψf (cid:0) − ( A + B ) bψ (cid:1) . Remark 6.
Exact solutions of the nonlinear hyperbolic equation u tt = au xx + uf ( u x − bu ) are constructed in the same way. ◦ . Exact solutions of some nonlinear PDEs can be obtained using theproposition below. Proposition 6.
Let a nonlinear PDE have an exact solution involvingtrigonometric functions of the form u = F ( x, t, A cos( βx ) + B sin( βx ) , β ) , (75) where A , B , and β are free real parameters that are not included in the consid-ered equation. Then this equation also has the exact solution involving hyperbolicfunctions: u = F ( x, t, ¯ A cosh( λx ) + ¯ B sinh( λx ) , − λ ) , (76) where ¯ A , ¯ B , λ are free real parameters. The converse is also true: if an equationhas the exact solution (76) , then it also has the exact solution (75) . Solution (76) is obtained from (75) by renaming the parameters β = iλ , A = ¯ A , B = − i ¯ B , i = − . Example 24.
Consider the fourth-order nonlinear equation u y (∆ u ) x − u x (∆ u ) y = ν ∆∆ u, ∆ u = u xx + u yy , (77) o which the stationary Navier–Stokes equations are reduced in the planarcase [54].Equation (77) has the exact solution u ( x, y ) = [ ¯ A sinh( λx ) + ¯ B cosh( λx )] e − γy + νγ ( γ + λ ) x. Therefore, this equation also has the exact solution u ( x, y ) = [ A sin( βx ) + B cos( βx )] e − γy + νγ ( γ − β ) x. These solutions and other examples of this kind can be found in [13].
5. Using Solutions of Simpler Equations for ConstructSolutions to Complex Equations
Preliminary remarks.
It is often possible to use solutions of simpler equationsto construct exact solutions to complex differential equations. In this section, wewill illustrate the reasoning in such cases for nonlinear PDEs (see Subsection 5.1),as well as for more complex nonlinear partial functional differential equations (seeSubsections 5.2–5.4).
The following example shows how precise solutions of nonlinear reaction-diffusionequations can be used to generate exact solutions to wave type equations.
Example 25.
Consider the reaction-diffusion equation with quadratic non-linearity u t = a ( uu x ) x + bu, (78)which admits several simple exact solutions, which are given below and expressedin elementary functions (see, for example, [13]). ◦ . The additive separable solution: u = − b a x + ψ ( t ) , (79)where ψ ( t ) = C exp (cid:0) bt (cid:1) and C is an arbitrary constant. ◦ . The multiplicative separable solution: u = ψ ( t ) x , (80)where ψ ( t ) = − be bt (6 ae bt + C ) − and C is an arbitrary constant. ◦ . The generalized separable solution: u = ψ ( t ) x + ψ ( t ) , (81)where ψ ( t ) = − be bt (6 ae bt + C ) − and ψ ( t ) = C e bt (6 ae bt + C ) − / , and C and C are arbitrary constants. ◦ . The generalized separable solution: u = ψ ( t ) x + ψ ( t ) √ x, (82)where ψ ( t ) = − be bt (6 ae bt + C ) − and ψ ( t ) = C e bt (6 ae bt + C ) − / , and C and C are arbitrary constants.Let us now consider a nonlinear wave type equation with a quadratic non-linearity of the form u tt = a ( uu x ) x + bu. (83)Equations (78) and (83) differ only in the order of the derivative with respectto t in the left parts of the equations. Since the right-hand sides of these equationsinvolving derivatives with respect to x are the same, it is natural to assume thatthe power structure of solutions with respect to x of both equations will also bethe same, and only the functional factors that depend on t will change for differentpowers of x .In other words, we look for exact solutions of wave type PDE (83) in thesame form as solutions of reaction-diffusion PDE (78). As a result, we get thefollowing four exact solutions of PDE (83): ◦ . The additive separable solution of the form (79), where the function ψ = ψ ( t ) is described by the ODE: ψ ′′ tt = − bψ + bψ. ◦ . The multiplicative separable solution of the form (80), where the function ψ = ψ ( t ) is described by the ODE: ψ ′′ tt = 6 aψ + bψ. ◦ . The generalized separable solution of the form (81), where the functions ψ = ψ ( t ) and ψ = ψ ( t ) are described by the ODEs: ψ ′′ = 6 aψ + bψ ,ψ ′′ = 2 aψ ψ + bψ . ◦ . The generalized separable solution of the form (82), where the functions ψ = ψ ( t ) and ψ = ψ ( t ) are described by the ODEs: ψ ′′ = 6 aψ + bψ ,ψ ′′ = aψ ψ + bψ . The considered example is a good illustration of a rather general fact, whichis a consequence of the results of [15] (see also [13, 14]) and can be formulated asthe following proposition.
Proposition 7.
Let the evolution partial differential equation u t = F [ u ] , (84) where F [ u ] ≡ F ( u, u x , . . . , u ( n ) x ) is the nonlinear differential operator in x , has ageneralized separable solution of the form u = m X k =1 ψ k ( t ) ϕ k ( x ) . (85) Then, the more complex partial differential equation L [ u ] = L [ w ] , w = F [ u ] , (86) where L and L are any linear differential operators in t , L [ u ] = k X i =0 a i ( t ) u ( i ) t , L [ w ] = m X j =0 b j ( t ) w ( j ) t , also has the generalized separable solution of the form (85) with the same functions ϕ k ( x ) (but with other functions ψ ( t ) ). Remark 7.
In the equations (84) and (86), the nonlinear operator F canexplicitly depend on the variables x and t . et us now give an example of constructing an exact solution that cannotbe obtained by using Proposition 7. Example 26.
Consider the n th-order nonlinear PDE: u tt = uF ( u x /u, u xx /u, . . . , u ( n ) x /u ) , (87)which differs from (32) only in the order of the derivative with respect to t on theleft part of the equation.We look for the solution of equation (87) in the same form as the solutionof equation (32). Substituting (29) in (87), for the function ϕ = ϕ ( z ) we obtain anonlinear ODE: k ϕ + 2 kqϕ ′ z + q ϕ ′′ zz = ϕF ( pϕ ′ z /ϕ, p ϕ ′′ zz /ϕ, . . . , p n ϕ ( n ) z /ϕ ) . In biology, biophysics, biochemistry, chemistry, medicine, control theory, climatemodel theory, ecology, economics, and many other areas there are nonlinear sys-tems, the rate of change of parameters of which depends not only on the currentstate of the system at a given time, but also on the state system at some previoustime [59]. The differential equations that describe such processes, in addition tothe unknown function u = u ( x, t ) also include the function w = u ( x, t − τ ) , where t is time, τ > is the constant delay. In some cases, we consider situations wherethe delay depends on the time, τ = τ ( t ) .The presence of a delay significantly complicates the analysis of such equa-tions. Although nonlinear PDEs with constant delay allow solutions of the trav-eling wave type u = u ( z ) , where z = x + λt (see, for example, [59–62]), they donot allow self-similar solutions of the form u = t β ϕ ( xt λ ) , which often have simplerPDEs without delay.More complex than traveling wave solutions, exact solutions of nonlinearreaction-diffusion type equations with delay were obtained in [63–72]. Exact solu-tions of nonlinear Klein–Gordon type equations with delay and related nonlinearhyperbolic equations are given in [71–76].Below, with specific examples, we will show how exact solutions of nonlineardelay PDEs can be found by using solutions of simpler PDEs without delay. xample 27. Let us consider a nonlinear reaction-diffusion equation witha constant delay, u t = a ( uu x ) x + bw, w = u ( x, t − τ ) . (88)Equation (88) is more complicated than the ODE without delay (78) andgoes into it at τ = 0 . The presence of the delay in (88) does not affect thenonlinear term containing derivatives in x . Therefore, we can assume that thepower structure of solutions in x of both equations will be the same, and only thefunctional factors that depend on t will change.In other words, we look for exact solutions PDE with delay (88) in thesame form, as solutions simpler PDE without delay (78). As a result, we get thefollowing four exact solutions of the nonlinear delay PDE (88): ◦ . The additive separable solution of the form (79), where the function ψ = ψ ( t ) is described by the linear delay ODE: ψ ′ t = − bψ + b ¯ ψ, ¯ ψ = ψ ( t − τ ) . ◦ . The multiplicative separable solution of the form (80), where the function ψ = ψ ( t ) is described by the nonlinear delay ODE: ψ ′ t = 6 aψ + b ¯ ψ, ¯ ψ = ψ ( t − τ ) . ◦ . The generalized separable solution of the form (81), where the functions ψ = ψ ( t ) and ψ = ψ ( t ) are described by the delay ODEs: ψ ′ = 6 aψ + b ¯ ψ , ¯ ψ = ψ ( t − τ ) ,ψ ′ = 2 aψ ψ + b ¯ ψ , ¯ ψ = ψ ( t − τ ) . ◦ . The generalized separable solution of the form (82), where the functions ψ = ψ ( t ) and ψ = ψ ( t ) are described by the delay ODEs: ψ ′ = 6 aψ + b ¯ ψ , ¯ ψ = ψ ( t − τ ) ,ψ ′ = aψ ψ + b ¯ ψ , ¯ ψ = ψ ( t − τ ) . Example 28.
More complex than (88), nonlinear PDE with variable delay u t = a ( uu x ) x + bw, w = u ( x, t − τ ( t )) , here τ ( t ) is an arbitrary function, also admits four exact solutions of the form(79)–(82). Example 29.
The reaction-diffusion equation with logarithmic nonlinearity u t = au xx + u ( b ln u + c ) , (89)admits the exact functional separable solution [15]: u ( x, t ) = exp[ ψ ( t ) x + ψ ( t ) x + ψ ( t )] , (90)where the functions ψ n = ψ n ( t ) are described by the nonlinear system of ODEs: ψ ′ = 4 aψ + bψ ,ψ ′ = 4 aψ ψ + bψ ,ψ ′ = a ( ψ + 2 ψ ) + bψ + c. Let us now consider a more complex nonlinear reaction-diffusion equationwith a constant delay, u t = au xx + u ( b ln w + c ) , w = u ( x, t − τ ) . (91)PDE with delay (91) in the special case τ = 0 passes into the simpler PDEwithout delay (89). For τ = 0 the solution of delay PDE (91), as for equation (89),is sought in the form (90). As a result, for the functions ψ n = ψ n ( t ) , we obtainthe nonlinear system delay ODEs: ψ ′ = 4 aψ + b ¯ ψ , ¯ ψ = ψ ( t − τ ) ,ψ ′ = 4 aψ ψ + b ¯ ψ , ¯ ψ = ψ ( t − τ ) ,ψ ′ = a ( ψ + 2 ψ ) + b ¯ ψ + c, ¯ ψ = ψ ( t − τ ) . Example 30.
More complex than (91), nonlinear PDE with variable delay u t = au xx + u ( b ln w + c ) , w = u ( x, t − τ ( t )) , where τ ( t ) is an arbitrary function, also admits a solution with functional separa-tion of variables of the form (90) [64]. Remark 8.
In [71, 72], methods for constructing generalized traveling-wavesolutions for nonlinear PDEs with delay by using exact solutions more simplePDEs without delay were proposed. This method is suitable for constructingexact solutions in both explicit and implicit form. .3. Pantograph-Type Partial Differential Equations In this section, we will consider functional-differential equations with partialderivatives of the pantograph type, which in addition to the unknown u = u ( x, t ) , also contain the same functions with dilated or contracted arguments, w = u ( px, qt ) , where p and q are scaling parameters (for equations with variabledelay we have < p < , < q < ). Pantograph-type ODEs and PDEs are usedfor mathematical modeling of various processes in engineering [77], biology [78–82],astrophysics [83], electrodynamics [84], the theory of populations [85], number the-ory [86], stochastic games [87], graph theory [88], risk and queuing theories [89],the theory of neural networks [90].Below, with specific examples, it is shown that the exact solutions of nonlin-ear pantograph-type PDEs can be found by using simpler ‘ordinary’ PDEs, whichdo not contain the unknown function with dilated or contracted arguments. Example 31.
Let us first consider a pantograph-type reaction-diffusionequation with quadratic nonlinearity, u t = a ( uu x ) x + bw, w = u ( px, qt ) . (92)Equation (92) is more complicated than the ODE without argument scal-ing (78) and passes into it at p = q = 1 . The presence in (92) of dilation in w doesnot affect the nonlinear term containing derivatives with respect to x . Therefore,we can assume that the power structure of solutions in x of both equations willbe the same, and only the functional factors that depend on t will change (andfor the additive separate solution, a factor in x will change).In other words, we look for exact solutions of pantograph-type PDE (93) inthe same form as solutions of ‘ordinary’ simpler PDE (78). As a result, we get thefollowing four exact pantograph-type solutions (92): ◦ . The additive separable solution is u = − bp a x + ψ ( t ); ψ ′ t = − bp ψ + b ¯ ψ, ¯ ψ = ψ ( qt ) . ◦ . The multiplicative separable solution has the form (80), where the func-tion ψ = ψ ( t ) is described by the pantograph-type ODE: ψ ′ t = 6 aψ + bp ¯ ψ, ¯ ψ = ψ ( qt ) . ◦ . The generalized separable solution has the form (81), where the functions ψ = ψ ( t ) and ψ = ψ ( t ) are described by the pantograph-type ODEs: ψ ′ = 6 aψ + bp ¯ ψ , ¯ ψ = ψ ( qt ) ,ψ ′ = 2 aψ ψ + b ¯ ψ , ¯ ψ = ψ ( qt ) . ◦ . The generalized separable solution has the form (82), where the functions ψ = ψ ( t ) and ψ = ψ ( t ) are described by the pantograph-type ODEs: ψ ′ = 6 aψ + bp ¯ ψ , ¯ ψ = ψ ( qt ) ,ψ ′ = aψ ψ + b √ p ¯ ψ , ¯ ψ = ψ ( qt ) . Example 32.
The reaction-diffusion equation with power-law nonlinearity u t = au xx + bu k , (93)for k = 1 admits a self-similar solution [25]: u ( x, t ) = t − k U ( z ) , z = xt − / , (94)where the function U = U ( z ) is described by the nonlinear ODE: − k U − zU ′ z = aU ′′ zz + U k , Let us now consider a much more complex nonlinear partial functional-differential equation of the pantograph-type u t = au xx + bw k , w = u ( px, qt ) , (95)where p and q are free parameters ( p > , q > ). Parameter values < p < and < q < correspond to equations with proportional delay in two arguments.The functional-differential equation (95) in the special case p = q = 1 passesinto the ‘ordinary’ partial differential equation (93). For k = 1 the solution of thepantograph-type PDE (95), as for equation (93), is sought in the form (94). As aresult, for the function U = U ( z ) , we obtain a nonlinear ODE of the pantograph-type [91]: − k U − zU ′ z = aU ′′ zz + bq k − k W k , W = U ( sz ) , s = pq − / . (96) emark 9. Equation (95) with proportional delays for < p, q < in thespecial case p = q / has an exact solution expressed in terms of the solution ofthe ODE without delay (96) with s = 1 ; for p < q / , Eq. (95) reduces to thedelay ODE with s < ; and for p > q / , to the ODE with contracted argumentfor s > . Moreover, a solution of the ODE (95) for p, q > for appropriate valuesof the parameters p and q can also be expressed in terms of the solution of theODE with delay ( s < ), without delay ( s = 1 ), and with contracted argument( s > ). Example 33.
Let us now consider the reaction-diffusion equation withexponential nonlinearity u t = au xx + be λu , (97)which for λ = 0 admits the invariant solution [25]: u ( x, t ) = U ( z ) − λ ln t, z = xt − / , (98)where the function U = U ( z ) is described by the nonlinear ODE: − λ − zU ′ z = aU ′′ zz + be λU . Let us now consider a much more complex nonlinear functional-differentialequation of the pantograph-type u t = au xx + be λw , w = u ( px, qt ) , (99)where p and q are free parameters ( p > , q > ).Partial functional-differential equation (99) for p = q = 1 passes into the ‘or-dinary’ partial differential equation (97). For λ = 0 the solution of the pantograph-type equation (99), as for equation (97), is sought in the form (98). As a result, forthe function U = U ( z ) we obtain a nonlinear ODE of the pantograph-type [91]: − λ − zU ′ z = aU ′′ zz + bq e λW , W = U ( sz ) , s = pq − / . In the special case for p = q / this equation is a standard ODE (without dilatedor contracted arguments). xample 34. It is easy to show that the nonlinear Klein–Gordon typeequation u tt = au xx + u ( b ln u + c ) (100)allows the multiplicative separable solution u ( x, t ) = ϕ ( x ) ψ ( t ) . (101)More complicated than (100), the nonlinear pantograph-type PDE: u tt = au xx + u ( b ln w + c ) , w = u ( px, qt ) , (102)also has the multiplicative separable solution (101), where the functions ϕ = ϕ ( x ) and ψ = ψ ( t ) are described by the nonlinear pantograph-type ODEs: aϕ ′′ xx + ϕ ( b ¯ ϕ + c ) = 0 , ¯ ϕ = ϕ ( px ); ψ ′′ tt = bψ ln ¯ ψ, ¯ ψ = ψ ( qt ) . Example 35.
More complex than (102), the partial functional-differentialequation u tt = au xx + u ( b ln w + c ) , w = u ( ξ ( x ) , η ( t )) , (103)where ξ ( x ) and η ( t ) are arbitrary functions, also allows a solution with the sepa-ration of variables of the form (101).In particular, for ξ ( x ) = x − τ and η ( t ) = t − τ , where τ and τ aresome positive constants, equation (103) is a partial differential equation with twoconstant delays. Below, a rather general approach for constructing exact solutions of functionalpartial differential equations of the pantograph-type is formulated in the formfollowing principle.
The principle of analogy of solutions.
Structure of exact solutions topartial functional-differential equations of the form F ( u, w, u x , u t , u xx , u xt , u tt , . . . ) = 0 , w = u ( px, qt ) (104) ften (but not always) is determined by the structure of solutions to simpler partialdifferential equations: F ( u, u, u x , u t , u xx , u xt , u tt , . . . ) = 0 . (105)Equation (105) does not contain the unknown functions with dilated or con-tracted arguments; it is obtained from (104) by formally replacing w by u .The solutions discussed in Examples 29–32 were constructed by using theprinciple of analogy of solutions. Below are two more complex examples. Example 36.
Consider the pantograph-type reaction-diffusion equationwith power-law nonlinearities u t = au xx + bu m w k , w = u ( px, qt ) , (106)that is more complex than (95).Following the principle of analogy of solutions, we set w = u in Eq. (106).As a result, we arrive at the equation u t = au xx + bu m + k , which, after renaming m + k to k coincides with Eq. (93). Taking into accountthat the solution of equation (93) is determined by formula (94), the solution ofequation (106) (by renaming k to m + k in Eq. (94)) are sought in the form [91]: u ( x, t ) = t − m − k U ( z ) , z = xt − / , k = 1 − m. As a result, for the function U = U ( z ) we get the nonlinear pantograph-typeODE: aU ′′ zz + 12 zU ′ z − − m − k U + bq k − m − k U m W k = 0 ,W = U ( sz ) , s = pq − / . Example 37.
Let us now consider the pantograph-type reaction-diffusionequation with exponential nonlinearities u t = au xx + be µu + λw , w = u ( px, qt ) , (107) hat is more complex than (99). Following the principle of analogy of solutions,we set w = u in Eq. (107). As a result, we arrive at the equation u t = au xx + be ( µ + λ ) u , which, after renaming µ + λ by λ coincides with Eq. (96). Taking into accountthat the solution of equation (99) is determined by formula (98), the solution ofequation (107) (by renaming λ to µ + λ in Eq. (98)) is sought in the form u ( x, t ) = U ( z ) − µ + λ ln t, z = xt − / , µ = − λ. As a result, for the function U = U ( z ) we obtain the nonlinear pantograph-typeODE: aU ′′ zz + 12 zU ′ z + 1 µ + λ + bq − λµ + λ e µU + λW = 0 ,W = U ( sz ) , s = pq − / . Remark 10.
In [91], a number of exact solutions of nonlinear pantographPDEs of diffusion and wave types are obtained, which well confirm the principleof analogy of solutions for pantograph-type PDEs.
6. Brief Conclusions
A number of simple, but quite effective, methods for constructing exact solutionsof nonlinear partial differential equations that require a relatively small amount ofintermediate calculations are described. These methods are based on the followingtwo main ideas: (i) simple exact solutions can serve as the basis for constructingmore complex solutions of the considered equations, (ii) exact solutions of someequations can serve as a basis for constructing solutions of more complex equa-tions. The effectiveness of the proposed methods is illustrated by a large numberof specific examples of constructing exact solutions of nonlinear heat equations,reaction-diffusion equations, wave type equations, hydrodynamics equations andsome other PDEs. In addition to exact solutions to partial differential equations,some exact solutions to nonlinear delay PDEs and pantograph-type PDEs are alsodescribed. The principle of analogy of solutions is formulated, which allows us toconstructively find exact solutions to such partial functional-differential equations. ote that the simple methods and examples described in this article canbe used in courses of lectures on equations of mathematical physics, methodsof mathematical physics and partial differential equations for undergraduate andgraduate students of universities. References
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