Nonlinear second order inhomogeneous differential equations in one dimension
NNonlinear second order inhomogeneous differentialequations in one dimension
Yajnavalkya BhattacharyaJurij W. DarewychYork University, Toronto, Canada
Abstract
We study inhomogeneous nonlinear second-order differential equations inone dimension. The inhomogeneities can be point sources or continuoussource distributions. We consider second order differential equations of type φ (cid:48)(cid:48) ( x ) + V ( φ ( x )) = Q δ ( x ), where V ( φ ) is a continuous, differentiable, an-alytic function and Q δ ( x ) is a point source. In particular we study cubicfunctions of the form V ( φ ( x )) = A φ ( x ) + B φ ( x ). We show that Greenfunctions can be determined for modifications of such cubic equations, andthat such Green’s functions can be used to determine the solutions for caseswhere the point source is replaced by a continuous source distribution.
1. Introduction
There are not many nonlinear differential equations in one dimension forwhich exact analytic solutions can be obtained in terms of elementary func-tions. This is especially so for nonlinear equations with an inhomogeneoussource term. This is noted in a textbook by Olver [1]: “....the Superposition Principle for inhomogeneous linear equations allowsone to combine the responses of the system to different external forcing func-tions...The two general Superposition Principles furnish us with powerfultools for solving linear partial differential equations, which we shall repeat-edly exploit throughout this text. In contrast, nonlinear partial differentialequations are much tougher, and, typically, knowledge of several solutions isof scant help in constructing others. Indeed, finding even one solution to anonlinear partial differential equation can be quite a challenge.”
Preprint submitted to Elsevier June 11, 2020 a r X i v : . [ m a t h . G M ] J un xact solutions to one-dimensional homogeneous nonlinear oscillators,such as the “restricted Duffing” equation d ydt + y + (cid:15)y = 0 where (cid:15) > , y (0) = A, dy (0) dt = 0 . (1)have been worked out in terms of Jacobi elliptic functions by Mickens [2], [3].Approximate Green’s functions for second order inhomogeneous nonlin-ear equations have been worked out by Frasca and Khurshudyan [4], andby others. For the nonrelativistic quartic oscillator, Anderson has derivedthe quantum mechanical Green’s function G ( z , t ; z , t ) [5], based on aninvertible linearization map [6].An analytic solution to the nonlinear (second-order) one-dimensional, cu-bic equation (cid:18) − d dx − µ (cid:19) φ ( x ) + λ φ ( x ) = 0 , (2)where µ and λ are constants, is given by [7] φ ( x ) = µ √ λ tanh (cid:18) xµ √ (cid:19) , (3)as can be simply verified by substituting the function (3) into eq. (2).
2. Inhomogeneous equations
It turns out that the inhomogeneous generalisation of (2), that is, thenonlinear cubic eq. (2) but with a point source on the right-hand side,namely (cid:18) − d dx − µ (cid:19) F ( x ) + λ F ( x ) = δ ( x ) , (4)can be solved analytically. This nonlinear, inhomogeneous eq. (4) has theanalytic solution: F ( x ) = µ √ λ tanh (cid:18) | x | µ √ (cid:19) , (5)2his can be verified by substituting eq.(5) into eq.(4). Note that | x | standsfor √ x , where, in one dimension, −∞ < x < + ∞ . Note also that thesolution F ( x ) of equation (5) is just F ( x ) = φ ( | x | ), where φ is given in eq.(3).We plot the solutions (5) in units of µ for a few values of λ in figure 1: λ = λ = λ = λ = - - - - F ( x ) Figure 1: Plot of F ( x ) = µ √ λ tanh (cid:18) | x | µ √ (cid:19) in units of µ =1, for various λ If µ in equation (4) is replaced by i m , where m is also real like µ , then themodified equation, (cid:18) − d dx + m (cid:19) ψ ( x ) + λ ψ ( x ) = 0 , (6)has the solution ψ ( x ) = m √ λ tan (cid:18) x m √ (cid:19) (7)3hus, the inhomogeneous version of eq. (6), (cid:18) − d dx + m (cid:19) Ψ( x ) + λ Ψ ( x ) = δ ( x ) , (8)has the solution Ψ( x ) = m √ λ tan (cid:18) | x | m √ (cid:19) , (9)which is plotted in figure 2. λ = λ = λ = λ = - - - - - - - - x Ψ( x ) Figure 2: Plot of Ψ( x ) = µ √ λ tan (cid:18) | x | m √ (cid:19) in units of m =1, for various λ
3. Generalisations
We can generalise the above results for second-order equations to anyequation of the form − d φ ( x ) dx + V (cid:0) φ ( x ) (cid:1) = 0 (10)4here V ( φ ) is a continuous differentiable analytic function. If the solutionof eq. (10) is φ ( x ), then the solution of the corresponding inhomogeneousequation with a point source, namely − d Φ( x ) dx + V (cid:0) Φ( x ) (cid:1) = δ ( x ) (11)will be Φ( x ) = φ ( | x | ). The second-order nonlinear equations (4) and (8),as well as their solutions (5) and (9), are particular examples of the generalresults (10) and (11).A particularly simple form of (11) is the linear one-dimensional neutrondiffusion equation with a plane source: − d φ ( x ) dx + k φ ( x ) = δ ( x ) . (12)Eq. (12) corresponds to equation (11) with V ( x ) = k φ ( x ). It has thesolution φ ( x ) = 12 k e −| kx | , (13)as can be verified by substituting eq. (13) into eq. (12).Regardless of whether φ ( x ) of eq. (10) is analytic, or can only be workedout numerically, it will nevertheless be true that the solution of (11) will be φ ( | x | ).
4. Green’s functions for the nonlinear equations and continuoussource distributions
Green’s function for the linear equation (12) can be obtained by replacingthe single point source δ ( x ) with a source of the form δ ( x − x ). Thus, thelinear equation (12) becomes the equation − d φ ( x ) dx + k φ ( x ) = δ ( x − x ) , (14)which has the solution φ ( x, x ; k ) = 12 k e − k | x − x | , (15)5xcept for x = x where it is singular. The result (15) is recognized as theGreen funtion for the Modified Helmholtz Equation in one dimension.[11]The analogous equation for the nonlinear case, cf. eq. (4), is − d G ( x ) dx − µ G ( x ) + λ G ( x ) = δ ( x − x ) . (16)The solution, i.e. the Green function for the nonlinear eq. (16), is G ( x, x ; µ, λ ) = µ √ λ tanh (cid:18) | x − x | µ √ (cid:19) , (17)except at x = x where G is singular.Analogously to (16), eq. (8) becomes (cid:18) − d dx + m (cid:19) Ψ( x ) + λ Ψ ( x ) = δ ( x − x ) , (18)which has the solution, i.e. Green function,Ψ( x, x ; m, λ ) = m √ λ tan (cid:18) | x − x | m √ (cid:19) . (19)Once the Green functions are known, it is possible to calculate the “po-tential” V ( x ) due to a continuous source (“charge”) distribution R ( x ), byevaluating the integral: V ( x ) = (cid:90) ∞−∞ dx R ( x ) G ( x, x ) . (20)For the relatively simple linear case, cf. eqs (14) and (15), V ( x ) can be eval-uated analytically for various R ( x ). For a Gaussian source distribution R ( x ) = e − x , the potential V l in ( x ) V l in ( x ) = (cid:90) ∞−∞ dx e − x k e − k | x − x | , (21)evaluates to V l in ( x ) = √ π k e (cid:16) − kx + k (cid:17) (cid:20) e kx Er (cid:18) x + k (cid:19) − e kx + Er (cid:18) − x + k (cid:19) − (cid:21) , (22)6here Er ( u ) = 2 √ π (cid:90) u e − t dt is the Error function. Figure 3: Potential V l in ( x ) = (cid:90) ∞−∞ dx e − x k e − k | x − x | .However, we are primarily concerned with the more interesting nonlinearcases (cf. eqs. (16) and (18)). Unfortunately, the evaluation of V ( x ), eq.(20), for a given distribution R ( x ), with either Green function (17) or (19),generally must be done by numerical quadrature.Analytic expressions for the pontential can be evaluated in a few cases;for example, a step-function source distribution, such as R ( x ) = 1 ∀ {− ≤ x ≤ } , R = 0 ∀{ x > , x < − } , (23)for which the potential is given by V ( x ) = (cid:90) ∞−∞ dx R ( x ) G ( x, x , ,
1) (24)7he integral in eq. (24), which can be evaluated analytically by Maple R (cid:13) andMathematica R (cid:13) , is plotted in Figure 4. The result is listed in Appendix I. Figure 4: Plot of V ( x ) of eq. (24), for the “step function” distribution R ( x ) of (23). Two other distributions for which analytic solutions of eq. (20) can be ob-tained are the “exponential” distribution R e ( x ) = e −| x | , and the Gaussian dis-tribution R g ( x ) = e − x . The resulting potentials V e ( x ) = (cid:90) ∞−∞ dx R e ( x ) G ( x, x , , V g ( x ) = (cid:90) ∞−∞ dx R g ( x ) G ( x, x , , V e ( x ) is listed in Appendix I. Numerical inte-grations of the potentials yield identical results to their analytic counterparts,as should be expected. V e ( x ) is plotted in figure 5:8 - Figure 5: Plot of V e ( x ) = (cid:90) ∞−∞ dx R e ( x ) G ( x, x , ,
1) , for the “exponential” distribution R e ( x ) = e −| x | . The plot of the potential V g ( x ) for the Gaussian distribution e − x turnsout to be very similar in shape, and, thus, it is not plotted here. Next, we exhibit examples of numerically calculated potentials V ( x ) withthe Green function of eq. (17), for a few choices of source distributions,namely R ( x ) = e − x /a (25) R ( x ) = 1[( x + a ) + b ) ] (26)where a and b are arbitrary, real constants. The potentials are V j ( x ) = (cid:90) ∞−∞ dx R j ( x ) G ( x, x , , , (27)where j = 2 , µ = λ = 1, and with a = b = 1), are plotted in Figure 6. 9 a) (b) Figure 6: (a) Potential V ( x ) of equation (27), with the Green function µ √ λ tanh( | x − x | µ √ ), µ = λ = 1, convolved with the Gaussian distribution e − x /a , a = 0 .
01; and(b) Potential V ( x ) of eq. (27), with the same Green function as in (a), µ = λ = 1,convolved with the distribution x + a ) + b ) ] , a = b = 1. The Green function given in eq. (19) is quite different from that of eq.(17), in that it has an infinity of singular points (vertical asymptotes), asindicated in figure 7aNevertheless it is possible to work out potentials in specified segmentsbetween adjacent vertical asymptotes. For example, if we consider a step-function distribution R ( x ) = 1 for 0 ≤ x ≤ R = 0; thenthe potential can be worked out analytically. Its plot is shown in figure 7b.10 a) (b) Figure 7: (a) Green function Ψ( x, x ; m, λ ) = m √ λ tan (cid:16) | x − x | m √ (cid:17) , for x = 1, m = λ = 1and (b) Potential V ( x ) of the Green function Ψ( x, x ; m = 1 , λ = 1) convolved with the“step function” distribution R ( x ) = 1 ∀ ≤ x ≤ µ = λ = 1. For the Gaussian or bell-shaped distributions given in eqs. (25) and(26) the potentials are as in eq. (20) but with the Green function as givenin eq. (19). In these cases the integrations must be done numerically inspecified segments. We plot two examples of such numerical determinationof potentials in Figure 8: 11 a) (b)
Figure 8: (a) Potential from the Green function m √ λ tan (cid:16) | x − x | m √ (cid:17) , m = λ = 1, convolvedwith the bell shaped distribution R (eq. 26), with a = b = 1, and(b) Potential from the Green function m √ λ tan (cid:16) | x − x | m √ (cid:17) , m = λ = 1, convolved with theGaussian distribution R (eq. 25) with a = 1.
5. Concluding remarks
We have shown that non-linear second-order equations in one dimensionfor which exact, analytic solutions can be obtained (cf. eqs. (2) and (6))can be generalised to inhomogeneous equations with delta function “point”sources for which Green’s functions can be determined (see eqs. (17) and(19)). If the sources are not point sources (“point charges”) but continuoussource distributions, then integral summations, that is “potentials”, can bedefined (cf. eq. (20)). These can be evaluated analytically in some cases, andotherwise by numerical quadrature, as discussed and illustrated in section 4.
6. Appendix I
The potential V ( x ) = (cid:90) ∞−∞ dx R ( x ) G ( x, x , µ = 1 , λ = 1), for the step-function R ( x ) = 1 ∀ {− ≤ x ≤ } , R = 0 ∀{ x > , x < − } R (cid:13) is shown in (28). V ( x ) = √ (cid:32) e √ x − + 1 e √ x +4) + 1 (cid:33) + 12 x ≤ − √ (cid:32) e √ x +4) + 1 e √ x − + 1 (cid:33) − x ≥ √ (cid:20) (cid:16) e √ x − + 1 (cid:17) (cid:16) e √ x +4) + 1 (cid:17)(cid:21) − x − − ≤ x ≤ V e ( x ) = (cid:90) ∞−∞ dx R e ( x ) G ( x, x , ,
1) for the exponential distribution R e ( x ) = e −| x | is evaluated in Mathematica in terms of Hypergeometric, and Polygammafunctions, shown in eq. 29. 13 e ( x ) = − e −√ x √ (cid:20) √ e √ x − √ e √ x + x + 4 e √ x − e √ x + x − e √ x F (cid:16) , − √ ; 1 − √ ; − e √ x (cid:17) − √ e √ x F (cid:16) , − √ ; 1 − √ ; − e √ x (cid:17) − e √ x F (cid:16) , √ ; 1 + √ ; − e −√ x (cid:17) −√ e √ x F (cid:16) , √ ; 1 + √ ; − e −√ x (cid:17) − e √ x F (cid:16) , √ ; 1 + √ ; − e √ x (cid:17) −√ e √ x F (cid:16) , √ ; 1 + √ ; − e √ x (cid:17) + √ F (cid:16) , √ ; 2 + √ ; − e −√ x (cid:17) + √ e √ x F (cid:16) , √ ; 2 + √ ; − e √ x (cid:17) +2 e √ x + x ψ (0) (cid:16) − √ (cid:17) + − √ e √ x + x ψ (0) (cid:0) (cid:0) − √ (cid:1)(cid:1) − e √ x + x ψ (0) (cid:0) (cid:0) − √ (cid:1)(cid:1) + 2 √ e √ x + x ψ (0) (cid:16) − √ (cid:17) (cid:21) x < e −√ x − x √ (cid:20) − e √ x − √ e √ x + 2 e √ x + x F (cid:16) , √ ; 1 + √ ; − e −√ x (cid:17) + √ e √ x + x F (cid:16) , √ ; 1 + √ ; − e −√ x (cid:17) −√ e x F (cid:16) , √ ; 2 + √ ; − e −√ x (cid:17) − e √ x ψ (0) (cid:16) √ (cid:17) − √ e √ x ψ (0) (cid:16) √ (cid:17) + e √ x ψ (0) (cid:0) (cid:0) √ (cid:1)(cid:1) + √ e √ x ψ (0) (cid:0) (cid:0) √ (cid:1)(cid:1) (cid:21) , x = 0 e −√ x − x √ (cid:20) − e √ x − √ e √ x + 2 e √ x + x F (cid:16) , √ ; 1 + √ ; − e −√ x (cid:17) + √ e √ x + x F (cid:16) , √ ; 1 + √ ; − e −√ x (cid:17) − e √ x + x F (cid:16) , √ ; 1 + √ ; − e √ x (cid:17) −√ e √ x + x F (cid:16) , √ ; 1 + √ ; − e √ x (cid:17) −√ e x F (cid:16) , √ ; 2 + √ ; − e −√ x (cid:17) + √ e √ x + x F (cid:16) , √ ; 2 + √ ; − e √ x (cid:17) − e √ x ψ (0) (cid:16) √ (cid:17) − √ e √ x ψ (0) (cid:16) √ (cid:17) +2 e √ x ψ (0) (cid:0) (cid:0) √ (cid:1)(cid:1) + 2 √ e √ x ψ (0) (cid:0) (cid:0) √ (cid:1)(cid:1) (cid:21) , x > eferences [1] Peter J. Olver Introduction to Partial Differential Equations , SpringerInternational Publishing, p. 12-13, 2014.[2] Ronald E Mickens
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