A new operational matrix technique to solve linear boundary value problems
AA NEW OPERATIONAL MATRIX TECHNIQUE TO SOLVE LINEARBOUNDARY VALUE PROBLEMS
UDAYA PRATAP SINGH ID DEPARTMENT OF APPLIED SCIENCESRAJKIYA ENGINEERING COLLEGE, SONBHADRA, UTTAR PRADESH, INDIA
EMAIL : [email protected]
ORCID ID : HTTPS://ORCID.ORG/0000-0002-4538-9377
Abstract.
A new technique is presented to solve a class of linear boundary value problems(BVP). Technique is primarily based on an operational matrix developed from a set of modifiedBernoulli polynomials. The new set of polynomials is an orthonormal set obtained with Gram-Schmidt orthogonalization applied to classical Bernoulli polynomials. The presented methodchanges a given linear BVP into a system of algebraic equations which is solved to find an ap-proximate solution of BVP in form of a polynomial of required degree. The technique is appliedto four problems and obtained approximate solutions are graphically compared to available ex-act and other numerical solutions. The method is simpler than many existing methods andprovides a high degree of accuracy.
Keywords: approximate solution of BVP; Bernoulli polynomials; boundary value problems;operational matrix; orthonormal polynomials.AMS Mathematics Subject Classification: 65L05; 34A45; 11B68 Introduction
Boundary value problems (BVP) have a lot of applications in areas of science, engineeringand technology. For illustration, rheological models, bio-fluid models, industrial engineering,hydrodynamics, lubrication problems, economics, ecology models, biological models, heat andmass transfer and many more are the examples where the BVPs naturally arise to play a sig-nificant role. It is often hard to find an analytic solution to these BVPs. In such situations,an approximate or numerical solution becomes an essential tool to deal with the problems. In-vestigations of numerical schemes to solve BVPs have been of concern from long past [1, 2, 3],however, in 20 th century, the advent of modern computers and software attracted much atten-tion of researchers towards high precision computations to numerical approximation problems[4, 5, 6], which has been of major concern in present times due to increasing demand of highprecision numerical solutions in different fields [7, 8, 9]. Some notable works on numerical orapproximate solutions of BVPs also include [10, 11, 12, 13]. Many authors used different poly-nomials such as Chebyshev polynomials [14], Legendre polynomials [15], Laguerre polynomialsand Wavelet Galerkin method [16], Legendre wavelets [4] to present various numerical schemes.Bernoulli polynomials and its properties have also been taken into account by many researchers[17, 18, 19, 20]. Recently, Singh et al. [20] used Bernoulli polynomials to solve Abel-Volterratype integral equations. However, numerical schemes always provide a numerical solution, itmay not qualify for further analytical applications in various situations. Therefore, the need ofa precise and simple approximate solution is always motivated.It is, therefore, proposed to solve linear boundary value problems of ordinary differentialequations using a class of modified Bernoulli polynomials and an operational matrix thereof tofind an approximate solution in the form of a polynomial. a r X i v : . [ c s . C E ] A ug UDAYA PRATAP SINGH ID DEPARTMENT OF APPLIED SCIENCES RAJKIYA ENGINEERING COLLEGE, SONBHADRA, UTTAR PRADESH, INDIA
EMAIL : [email protected]
ORCID ID : HTTPS://ORCID.ORG/0000-0002-4538-9377 Modified Bernoulli Polynomials
Classical Bernoulli polynomials are given as [21]:(1) B n ( ζ ) = n (cid:88) j =0 (cid:18) nj (cid:19) ´ b j ζ n − j , n = 0 , , , ... ≤ ζ ≤ b j are the Bernoulli numbers, which can be easily calculated with Kroneckers formula[22]:(2) B n (0) = − n +1 (cid:88) j =1 ( − j j (cid:18) n + 1 j (cid:19) j (cid:88) k =1 k n ; n ≥ B ( ζ ) =1 , B ( ζ ) = ζ − , B ( ζ ) = ζ − ζ + , B ( ζ ) = ζ − ζ + ζ, B ( ζ ) = ζ − ζ + ζ − .Many interesting properties of Bernoulli polynomials have been studied by different researchersfrom time to time [23, 24]. Two of its properties that are of interest in the present work are thatthese polynomials form a complete basis over [0 ,
1] [24], and their integral over [0 ,
1] is uniformlyzero [23],(3) (cid:90) B n ( z ) dz = 0 , n ≥ . Some other properties such as: B (cid:48) n ( ζ ) = nB n − ( ζ ) , n ≥ ,B n ( ζ + 1) − B n ( ζ ) = nζ n − , n ≥ Gram-Schmidt orthogonalization.
Property (3) shows that the polynomials B n ( ζ ) ( n ≥
1) (1) are orthogonal to B o ( ζ ) with respect to standard inner product on L ∈ [0 ,
1] defined as:(5) < f , f > = (cid:90) f ( x ) ¯ f ( x ) dx ; f , f ∈ L [0 , n + 1 polynomials is derived for any B n with Gram-Schmidt orthogonalization. For illustration, n = 5 gives following set of modifiedorthonormal polynomials: φ ( ζ ) = 1 φ ( ζ ) = √ − ζ ) φ ( ζ ) = √ (cid:0) − ζ + 6 ζ (cid:1) φ ( ζ ) = √ − ζ − ζ + 20 ζ ) φ ( ζ ) = 3(1 − ζ + 90 ζ − ζ + 70 ζ ) φ ( ζ ) = √ − ζ − ζ + 560 ζ − ζ + 252 ζ )(6)2.2. Operational matrix.
On integration over the interval [0 , (cid:90) ζ φ o ( η ) dη = 12 φ o ( ζ ) + 12 √ φ ( ζ )(8) ζ (cid:82) φ i ( x ) dx = √ (2 i − i +1) φ i − ( ζ )+ √ (2 i +1)(2 i +3) φ i +1 ( ζ ) , ( f or i = 1 , , ... , n ) NEW OPERATIONAL MATRIX TECHNIQUE TO SOLVE LINEAR BOUNDARY VALUE PROBLEMS 3 which can be represented in following closed form:(9) ζ (cid:90) φ ( η ) dη = Θ φ ( ζ )where ζ ∈ [0 ,
1] and Θ is operational matrix of order ( n + 1) given as :(10) Θ = 12 √ . · · · − √ . √ . · · · − √ . √ (2 n − n +1) · · · − √ (2 n − n +1) Solution of Boundary Value Problems
Approximation of Functions.Theorem 3.1.
Let H = L [0 , be a Hilbert space and Y = span { y , y , y , ..., y n } be a subspaceof H such that dim ( Y ) < ∞ , every f ∈ H has a unique best approximation out of Y [24] ,that is, ∀ y ( t ) ∈ Y, ∃ ˆ f ( t ) ∈ Y s.t. (cid:107) f ( t ) − ˆ f ( t ) (cid:107) ≤(cid:107) f ( t ) − y ( t ) (cid:107) . This implies that, ∀ y ( t ) ∈ Y, < f ( t ) − f ( t ) , y ( t ) > = 0 , where <, > is standard inner product on L ∈ [0 , (c.f.Theorems 6.1-1 and 6.2-5, Chapter 6 [24] ). Remark 3.2.
Let Y = span { φ , φ , φ , ..., φ n } , where φ k ∈ L [0 , are orthonormal Bernoullipolynomials. Then, from Theorem 3.1, for any function f ∈ L [0 , , (11) f ≈ ˆ f = n (cid:88) k =0 c k φ k , where c k = (cid:104) f, φ k (cid:105) , and <, > is the standard inner product on L ∈ [0 , as defined by equation(5). For numerical approximation, series (11) can be written as:(12) f ( ζ ) (cid:39) n (cid:88) k =0 c k φ k = C T φ ( ζ )where C = ( c , c , c , ..., c n ) , φ ( ζ ) = ( φ , φ , φ , ..., φ n ) are column vectors, and number of poly-nomials n can be chosen to meet required accuracy.3.2. Scheme of Approximation.
In order to present the basic ingredients of the method insimpler way, general form of second order linear ordinary differential equation with constantcoefficients will be considered first; and application of the method to higher order BVPs ofsimilar kind will be discussed subsequently.Let us consider the linear ordinary differential equation with constant coefficients:(13) d ydζ + a dydζ + a y = r ( ζ )Without loss of generality, we assume that the ODE (13) satisfy the boundary conditions(BCs):(14) y (0) = α, y (1) = β for if the boundary conditions be y ( ζ ) = α, y ( ζ ) = β , BCs (14) can be attained with thetransformation ζ → ζ − ζ ζ − ζ . It is further assumed that y and r are continuous functions of ζ ∈ [0 ,
1] and BVP (13-14) admits a unique solution on [0 , UDAYA PRATAP SINGH ID DEPARTMENT OF APPLIED SCIENCES RAJKIYA ENGINEERING COLLEGE, SONBHADRA, UTTAR PRADESH, INDIA
EMAIL : [email protected]
ORCID ID : HTTPS://ORCID.ORG/0000-0002-4538-9377
Let C = ( c o , c , c , ..., c n ) be a column vector of n + 1 unknown quantities such that(15) d ydζ = C T φ ( ζ )Equations (15) and (9-10) give: dydζ = β − α − C T Θ φ (1) + C T Θ φ ( ζ ) y ( ζ ) = α + ( β − α − C T Θ φ (1)) ζ + C T Θ φ ( ζ )(16)where, φ (1) = (cid:0) , √ , √ , . . . , √ n + 1 (cid:1) T .Substituting equations (15-16) into ODE (13), we get:(17) C T (cid:0) I + a Θ + a Θ (cid:1) φ ( ζ ) − C T ( a ζ + a ) Θ φ (1) = r ( ζ )where, I is identity matrix of order n + 1. Again, writing(18) ( a ζ + a ) Θ φ (1) = L φ ( ζ )and(19) r ( ζ ) − ( β − α )( a ζ + a ) − a α = R T φ ( ζ ) , equation (17) is simplified to following form:(20) C T (cid:0) I + a Θ + a Θ + L (cid:1) φ ( ζ ) = R T φ ( ζ )where, R = ( r o , r , ..., r n ) is a real column vector and L (for this case) is calculated as,(21) L = ( a + 2 a ) √ a o · · · − √ ( a + 2a ) − a · · ·
00 0 0 · · · · · · ( n +1) × ( n +1) From equation (20) and (15), the unknown coefficient C T and approximate solution to BVP(13-14) are obtained as:(22) C T = R T (cid:0) I + a Θ + a Θ + L (cid:1) − (23) y ( ζ ) = α + (cid:0) β − α − C T Θ φ (1) (cid:1) ζ + C T Θ φ ( ζ ) . Remark 3.3.
For the shake of completeness, let the BVP under consideration be of order n .Following obvious management will be required to the intermediate steps: • n th derivative will be set to C T φ (say, d n ydζ n = C T φ ). • left hand side of equation (19) will be adjusted with the polynomial generated due to n number of BC S . • equation (22) will take the form C T = R T (cid:0) I + a n − Θ + a n − Θ + · · · + a Θ n + L (cid:1) − with appropriately calculated L . 4. Examples
In this section, four examples have been considered to demonstrate the efficacy of the method.The first example is taken to demonstrate the scheme of approximation, second and fourthexamples are taken from published investigations, and the third example is selected for thereason that it has no easy analytic solution.
NEW OPERATIONAL MATRIX TECHNIQUE TO SOLVE LINEAR BOUNDARY VALUE PROBLEMS 5
Example 4.1.
As a first example, let us consider the following simple boundary value problem: d y dx − dydx + 6 y = e − x ; y (0) = 0 , y (1) = 5(24) which has exact solution y ( x ) = (cid:18) e − x − ( e +60 e − ) e ( e − e x + ( e +60 e − ) e ( e − e x (cid:19) . Comparing equation (24) with equation (13) for n = 7 ans 10, equations (22 - 23) yield C Tn =7 = (18 . , . , . , . , . , . , . , . C Tn =10 = (18 . , . , . , . , . , . , . , . , . , . × − , . × − (cid:1) (25) y ( x ) n =7 ≈ . x + 0 . x + 1 . x + 0 . x + 2 . x − . x + 0 . x y ( x ) n =10 ≈ − . × − + 0 . x + 0 . x + 1 . x + 1 . x +1 . x − . x + 0 . x − . x +0 . x − . x (26)A comparison of approximations (26) with exact solution of IVP (24) is shown in figure 1.Maximum magnitude of the error between exact and present solutions is of order 10 − and 10 − for n = 7 and n = 10, respectively. It is notable that the error for n = 10 is equivalent to errorof concatenated series of exact solution at degree 15.(a)(b) Figure 1. (a) Comparison of exact and present solution for example 4.1 for n = 7 ,
10. (b) Absolute error between exact and approximate solutions of 4.1 for n = 7 and 10. UDAYA PRATAP SINGH ID DEPARTMENT OF APPLIED SCIENCES RAJKIYA ENGINEERING COLLEGE, SONBHADRA, UTTAR PRADESH, INDIA
EMAIL : [email protected]
ORCID ID : HTTPS://ORCID.ORG/0000-0002-4538-9377
Example 4.2.
Let us consider the following boundary value problem of order : d y dx − y = − e x y ( k ) (0) = 0 , k = 0 , , , , y ( k ) (1) = − k e, k = 0 , , , where, y ( k ) ( x ) = d k y dx k . This BVP admits the exact solution y ( x ) = (1 − x ) e x . Wazwaz [26] presented an approximate solution of BVP (27) as polynomial of degree 12 andgot an error of order 10 − . In order to compare our solution to that by Wazwazz [26], we willpresent an approximation of degree 7 and 12.(a) (b)(c) (d) Figure 2. (a) Comparison of exact and present solution to example 4.2 for n = 7 ,
12. (b) Absolute error between exact and approximate solutions for n = 7(present approximation). (c) Absolute error between exact and approximatesolutions [26]. (d) Absolute error between exact and approximate for n = 12(present approximation).Proceeding with the BVP (27) as discussed in section 3.2 for n ( n = 7 , C Tn =7 = ( − . , − . , − . , − . , − . , − . , − . × − , − . × − (cid:1) C Tn =12 = ( − . , − . , − . , − . , − . , − . , − . , − . × − , − . × − , − . × − , − . × − , − . × − , − . × − (cid:1) (28) NEW OPERATIONAL MATRIX TECHNIQUE TO SOLVE LINEAR BOUNDARY VALUE PROBLEMS 7 y ( x ) n =7 ≈ − . × − x − . x − . x − . x − . x − . x − . x − . x y ( x ) n =12 ≈ − . × − + 0 . x + 0 . x + 1 . x + 1 . x +1 . x − . x + 0 . x − . x + 0 . x − . x − . × − x − . × − x − , − for n = 7 and n = 12, respectively. It is notable that the error of approximationof 12 th degree polynomial by Wazwaz [26] is of order 10 − , which is closer to that for n = 7 ofpresent solution, whilst our solution for n = 12 is much more accurate than Wazwaz [26]. Example 4.3.
Consider the ODE (30) d ydx − dydx + 2 y = tan( x ); y (0) = (cid:18) dydx (cid:19) x =0 = 0 which is linear in nature but its not easy to solve manually in terms of simply known mathemat-ical functions. We will compare the present solution of this problem with numerical solutionsgenerated by Mathematica. Proceeding as in previous examples for n = 9 ,
11, we get(31) C T = (5 . , . , . , . , . , . , . , . , . y ( x ) ≈ . x − . x + 0 . x + 0 . x + 0 . x − . x +1 . x − . x + 0 . x (32)The approximate solution (32) is compared with exact solution of IVP (30) and observedabsolute error of orders 10 − , − for n = 9 ,
11, respectively (Figure 2).(a) (b)
Figure 3. (a) Comparison of present approximation and
M athematica gener-ated numerical solutions to example 4.2 for n = 9 ,
11. (b) Absolute error betweenpresent approximation and
M athematica generated numerical solutions for n =9 , Example 4.4.
As a last example, let us take the following BVP [27] of order four: d ydx − d ydx − y = ( x − e x y (0) = (1) , y (1) = 0 , (cid:18) dydx (cid:19) x =0 = 0 , (cid:18) dydx (cid:19) x =1 = − e (33) which is admits the exact solution y ( x ) = (1 − x ) e x . UDAYA PRATAP SINGH ID DEPARTMENT OF APPLIED SCIENCES RAJKIYA ENGINEERING COLLEGE, SONBHADRA, UTTAR PRADESH, INDIA
EMAIL : [email protected]
ORCID ID : HTTPS://ORCID.ORG/0000-0002-4538-9377
Barari et al. [27] presented an approximate solution of degree 11 with variational iterationmethod (VIM) and got an error of order 10 − . We have presented solutions for n = 7 ,
10 andobtained errors of order 10 − and 10 − , respectively.Proceeding as in previous examples for n = 7 ,
10, we obtained C T ( n =7) = ( 0 . , − . , − . , − . , − . , − . , − . × − , − . × − ) C T ( n =10) = ( 0 . , − . , − . , − . , − . , − . , − . × − , − . × − , − . × − , − . × − , . × − )(34) y ( x ) ( n =7) ≈ . − . x − . x − . x − . x − . x − . x y ( x ) ( n =10) ≈ . − . x − . x − . x − . x − . x − . x − . x − . x − . × − x (35)Figure 4 shows comparison of present approximation with exact solution to example 4.4 for n = 7 ,
9. It is easy to observe that the present method for n = 7 yields similar error as in [27],but our solution for n = 10 yields far better approximation than that obtained in [27]. If valueof n is taken higher, more accurate solution will be obtained.(a)(b) (c) Figure 4. (a) Comparison of exact and present solution for example 4.4 for n = 7 ,
10. (b) Absolute error for n = 7 . (c) Absolute error for n = 10 . NEW OPERATIONAL MATRIX TECHNIQUE TO SOLVE LINEAR BOUNDARY VALUE PROBLEMS 9 Conclusion
A new scheme was presented and demonstrated to approximate the solution of linear boundaryvalue problems with constant coefficients. Gram-Schmidt orthogonalization and standard innerproduct of L [0 ,
1] applied to a set of first n Bernoulli polynomials produced a new class of n orthonormal polynomials showing a special tri-diagonal operational matrix, which were utilizedas a tool to transform a BVP into a system of algebraic equations with unknown coefficients.These unknown coefficients are evaluated with the scheme discussed in present method and,thereby, a polynomial approximation to the solution of the BVP is obtained. The method wasexplored with three examples. The main benefits of this method can be concluded as follows: • approximate solution comes out to be a polynomial of degree n , which enables the furtherapplication of solution. • approximation contain small errors, which can be minimized by considering higher degreeof Bernoulli polynomials. • method is fast in comparison to many available numerical and approximation methods. References [1] H. B. Keller,
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