Solving high-order partial differential equations in unbounded domains by means of double exponential second kind Chebyshev approximation

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1 Computational Methods for Differential quations Vol. 3, No. 3, 015, pp Solving high-order partial differential equations in unbounded domains by means of double exponential second kind Chebyshev approximation Mohamed Abdel-Latif Ramadan Mathematics Department, Faculty of Science, Menoufia niversity, Shebein l-koom, gypt. -mail: Kamal Mohamed Raslan Mathematics Department, Faculty of Science, Al-Azhar niversity, Nasr-City, Cairo, gypt. -mail: kamal Talaat l-sayed l-danaf Department of Mathematics and Statistics, Taibah niversity Madinah Munawwarah, KSA. -mail: Mohamed Ahmed Abd l Salam Mathematics Department, Faculty of Science, Al-Azhar niversity, Nasr-City, Cairo, gypt. -mail: mohamed salam1985@yahoo.com Abstract In this paper, a collocation method for solving high-order linear partial differential equations PDs with variable coefficients under more general form of conditions is presented. This method is based on the approximation of the truncated double exponential second kind Chebyshev SC series. The definition of the partial derivative is presented and derived as new operational matrices of derivatives. All principles and properties of the SC functions are derived and introduced by us as a new basis defined in the whole range. The method transforms the PDs and conditions into block matrix equations, which correspond to system of linear algebraic equations with unknown SC coefficients, by using SC collocation points. Combining these matrix equations and then solving the system yield the SC coefficients of the solution function. Numerical examples are included to test the validity and applicability of the method. Keywords. xponential second kind Chebyshev functions, High-order partial differential equations, Collocation method. 010 Mathematics Subject Classification. 65N04, 35G06. Received: 30 December 015 ; Accepted: 1 May 016. Corresponding author. 147

2 148 M. RAMADAN, K. RASLAN, T. L DANAF, AND M. ABD L SALAM 1. Introduction It is well known that the numerical methods have played an important role in solving PDs. Some of the well-known numerical methods are finite differences and finite element methods [6,17]. Recently, various approximate methods were discussed, such as differential transform method, Adomian decomposition method and Homotopy analysis method [3,8,11,13,15,18]. Furthermore, spectral methods are one of the principal methods for solving differential equations. The main idea of spectral methods is to approximate the solutions of differential equations by means of truncated series of orthogonal polynomials. The most used versions of spectral methods are tau, collocation, and Galerkin methods [1, 5, 9]. One of the most important orthogonal polynomials is Chebyshev polynomials. The well-known Chebyshev polynomials of the second kind n x [1] are orthogonal with respect to the weight-function wx = 1 x on the interval [ 1, 1], and the recurrence relation is 0 x = 1, 1 x = x, n+1 x = x n x n 1 x, n 1. Many studies of Chebyshev polynomials are considered on the first kind T n x. On the other hand, in [10] and [14] a modified type of Chebyshev polynomials was proposed as new alternative technique to the solutions of ordinary and partial differential equations given in whole domain. In their studies, the basis functions called exponential Chebyshev functions, which are orthogonal on, and are defined by e x 1 n x = T n e x, + 1 where T n x is the first kind Chebyshev polynomials. This kind of extension tackles the problems over the whole real domain. In this paper we introduce a new type of Chebyshev polynomials in the whole real range, called exponential second kind Chebyshev SC functions. The rest of the paper is organized as follows. In section, the definition and properties of SC functions are listed, while in section 3 the form of high-order linear non-homogeneous partial differential equations is presented. In section 4, we formulated the fundamental matrix relation based on collocation points. In section 5, method of solution is presented. Finally, section 6 contains numerical illustrations and results that are compared with the exact solutions to demonstrate the applicability and accuracy of the present method.

3 CMD Vol. 3, No. 3, 015, pp Properties of double SC functions We introduce the definition of expotennial second kind Chebyshev SC to be of the form e n x 1 x = n e x, where the corresponding recurrencs relation is e 0 x = 1, 1 x 1 x = e x, + 1 e n+1x x 1 = e x n x + 1 n 1x. In Basu [4], the expression T r,s x, y = T r x T s y has given which is a form of Chebyshev polynomials. Mason et al. [1] have also used double Chebyshev polynomials expression for an infinitely differentiable function ux, y defined on the square S < x, y <, where T r x and T s y are Chebyshev polynomials of the first kind. Now we employ our new definition n x to double form. Definition.1. The double SC functions are in the following form r,sx, y = r x s y,. and the recurrence relation takes the form { } r+1,sx, e y = x 1 e x +1 r x r 1x s y, r 1, { } r,s+1x, y = r x s y s 1 y, s 1. e y 1 e y Orthogonality of double SC functions. The functions r,sx, y are orthogonal with respect to the weight function 3 wx, y = 4e x+y e x e y + 1 3, with the orthognality condition π i,jx, yk,lx,, i = j and k = l, ywx, ydxdy = 4 0, otherwise..4

4 150 M. RAMADAN, K. RASLAN, T. L DANAF, AND M. ABD L SALAM Also the product relation of double SC functions used in the partial derivatives relations is given by e x 1 e x + 1 e y 1 e y + 1 m,nx, y = 1 4 [ m+1,n+1x, y + m+1,n 1x, y + m 1,n+1x, y + m 1,n 1x, y]... Function expansion in terms of double SC functions. A function ux, y well defined over the square S < x, y <, can be expanded as ux, y = a r,s r,sx, y,.5 where r=0 s=0 a r,s = 4 π ux, y r,sx, ywx, ydxdy. If ux, y in expression.5 is truncated to n, m < in terms of the double SC functions then, it will takes the form m n x, y = a r,s r,sx, y, = x, y A,.6 r=0 s=0 where x, y is 1 m+1n+1 vector with elements r,sx, y and A is an unknown coefficient column vector, where x, y = [ 0,0x, y 0,1x, y... 0,nx, y 1,0x, y 1,1x, y... 1,nx, y... m,0x, y m,1x, y... m,nx, y],.7 and A = [a 0,0 a 0,1... a 0,n a 1,0 a 1,1... a 1,n... a m,0 a m,1... a m,n ] T The partial derivatives of double SC functions. The operational matrices of derivatives of the double SC functions are given in the next proposition Proposition.. The relation between the row vector x, y and its i,jth-order partial derivative is given as i,j x, y = x, yd x i D y j,.9

5 r 1,sx, y 1,0 ],.14 CMD Vol. 3, No. 3, 015, pp where, D x and D y are the m + 1n + 1 m + 1n + 1 operational matrices for the partial derivatives, and the general form of them is and α D x = diag γ α I, 0, { α = 0, 1,..., m, γ α = η η 0 D y = η β = 0, 1,..., n, γ β = T T α 4 I, 0, if α = 0, 1, if α 0, β, η = diag γ β, 0, { 0, if β = 0, 1, if β 0, β, where I and 0 are n + 1 n + 1 identity and zero matrices in the block matrix D x which is m + 1 m + 1. Also η is the matrix of n + 1 n + 1 in the block matrix D y which is m + 1 m + 1. Proof. The partial derivatives of the SC functions can be obtained by differentiating relation.3, first with respect to the variable x, and with the help of.6 we get x 0,s x, y = 0, for all s,.1 and x 1,s x, y = 4ex 1+e x s y = x 1 4 x s y = 3 4 0,sx, y 1 4,sx, y, x r+1,s x, y = [ x 1,s x, yr,sx, y r 1,sx, y ] = [ x 1,sx, y 0,0 r,s x, y 0,0 r 1,s x, y ] 0,0 = [ 1,sx, y 1,0 r,s x, y 0,0 + 1,s x, y 0,0 r,s x, y 1,0.13

6 15 M. RAMADAN, K. RASLAN, T. L DANAF, AND M. ABD L SALAM by using the relations and with the help of product relation for r = 0, 1,..., m, the elements of the matrix of dersvatives D x can be obtained from the following equalities 0,s x, y 1,0 = 0, 1,s x, y 1,0 = 3 4 0,sx, y 1 4,sx, y,,s x, y 1,0 = 1,s x, y 1 3,sx, y,. r,s x, y 1,0 = r+ 4 r 1,sx, y r 4 r+1,sx, y, r > 1, for all s. Similarly, we get the partial derivative with respect to the variable y as r,0 x, y 0,1 = 0, r,1 x, y 0,1 = 3 4 r,0x, y 1 4 r,x, y, r, x, y 0,1 = r,1 x, y 1 r,3x, y,. r,s x, y 0,1 = s+ 4 r,s 1x, y s 4 r,s+1x, y, s > 1, for all r Then, the previous equalities.15,.16 form m + 1n + 1 m + 1n + 1 two operational matrices D x and D y by our consideration that r,s x, y 1,0 = r,s x, y 0,1 = r,s x, y 0,0 = 0, for r > m and s > n. Thus, to obtain the matrix i,j x, y in terms of x, y, we can use the relation.15,.16 as 1,0 x, y = x, yd x,,0 x, y = 1,0 x, yd x = x, yd x D x = x, y D x, 3,0 x, y = 1,0 x, y D x, = x, y D x 3. Therefore, by induction we can write i,0 x, y = x, y D x i,.17 similarly 0,j x, y = x, y D y j,.18

7 CMD Vol. 3, No. 3, 015, pp hence, by using.17,.18 we get finally i,j x, y = x, y D x i D y j,.19 which end the proof. 3. Application of the introduced partial derivatives for high-order PDs The forms of high-order linear non-homogeneous partial differential equations with variable coefficients in unbounded domains are p r q i,j x, y u i,j x, y = fx, y, < x, y <, 3.1 i=0 j=0 with the non-local conditions [7], [16] ρ p r b t i,ju i,j ω t, η t = λ, t=1 i=0 j=0 and / or ν p r c t i,jxu i,j x, γ t = gx, 3. t=1 i=0 j=0 and / or θ p r d t i,jy u i,j ε t, y = hy, t=1 i=0 j=0 where the u 0,0 x, y = ux, y, u i,j x, y = i+j x i y ux, y and q j i,j x, y, fx, y, c t i,j x, gx, dt i,j y and hy are known functions on the square S < x, y <,, and ω t, η t, γ t, ε t are constants, and may be one or more of them tends to infinity. Now, we consider that the approximate solution x, y to the exact solution ux, y of q. 3.1 defined by expression.6 and its i, j-th partial derivatives defined by q..19 as m n x, y = a r,s r,s x, y = x, y A, 3.3 r=0 s=0 and i,j x, y = [x, yd x i D y j ]A. 3.4

8 154 M. RAMADAN, K. RASLAN, T. L DANAF, AND M. ABD L SALAM 4. Fundamental matrix relations Let us define the collocation points [10, 14], so that < x i, y i <, as 1+cos x k = Ln kπ m 1+cos, y 1 cos kπ m l = Ln lπ n, 1 cos lπ n k = 1,..., m 1, l = 1,..., n 1 and at the boundaries 4.1 k = 0, k = m x 0, x m, l = 0, l = n y 0, y n. Since the double SC functions are convergent at both boundaries ±, then the appearance of infinity in the collocation points does not cause a loss or divergence in the method. Now, we substitute the collocation points 4.1 into q. 3.1 to obtain p r q i,j x k, y l u i,j x k, y l = fx k, y l, 4. i=0 j=0 the system 4. can be written in the matrix form p r Q i,j i,j = F, p m, r n, 4.3 i=0 j=0 where Q i,j denotes the diagonal matrix with inner elements are q i,j x k, y l where, k = 0, 1,,..., m ; l = 0, 1,,..., n and F denotes the column matrix with the elements fx k, y l where, k = 0, 1,,..., m ; l = 0, 1,,..., n, by substituting the collocation points 4.1 into derivatives of the unknown function as in q. 3.4 yields where i,j x 0, y 0. i,j x 0, y n i,j x i,j 1, y 0 = = [D x i D y j ]A, 4.4. i,j x 1, y n. i,j x n, y m = [x 0, y 0 x 0, y 1... x 0, y n x 1, y 0 x 1, y 1... x 1, y n... x m, y 0 x m, y 1... x m, y n ] T,

9 CMD Vol. 3, No. 3, 015, pp therefore, from q. 4.3, we get a system of matrix equation fundamental matrix for the PD in the following form p r { Q i,j x, ydx i D y j} A = F, 4.5 i=0 j=0 which corresponds to a system of m + 1n + 1 linear algebraic equations with m+1n+1 double SC coefficients a r,s unknowns. By substituting the collocation points 4.1 in the condition 3. by same procedure before we get the fundamental matrices for conditions as ρ p r t=1 i=0 j=0 bt i,j { ωt, η t D x i D y j} A = λ, ν p r t=1 i=0 j=0 ct i,j x k { x k, γ t D x i D y j} A = gx k, θ p r t=1 i=0 j=0 dt i,j y l { ε t, y l D x i D y j} A = hy l. 4.6 It is also noted that the structure of matrices Q i,j and F vary according to the number of collocation points and the structure of the problem. However,, D x and D y do not change their nature for fixed values of m and n which are truncation limits of the SC series. In the other words, the changes in, D x and D y are only dependent on the number of collocation points. 5. Method of solution The fundamental matrix 4.5 for q. 3.1 corresponding to a system of m+1n+ 1 algebraic equations for the m+1n+1 unknown SC coefficients a 0,0, a 0,1,, a 0,n, a 1,0, a 1,1,, a 1,n,, a m,0, a m,1,, a m,n. We can write the matrix 4.5 as WA=F, or [W; F], 5.1 and we can obtain the matrix form for the conditions by means of 4.6 in a compact form as VA = R, or [V; R], 5. where V is a h m + 1n + 1 matrix and R is a h 1 matrix, so that h is the rank of the all row matrices as in 4.6 belong to the given conditions. Then 5.1 together with 5. can be written in the following compact form: W A = F, or [W ; F ]. 5.3 Furthermore, the system 5.3 can be formed by appending the rows 5. on conditions to the system 5.1. Then the size of the system of algebraic equations increases

10 156 M. RAMADAN, K. RASLAN, T. L DANAF, AND M. ABD L SALAM and therefore W becomes a rectangular matrix. To solve this new system, the generalized inverse of W can be used [7], and so the double SC coefficients can be found as A = geninvw F. The method procedure can be summarized by the following algorithm: 1. Calculating the matrix W;. Forming the matrix W by adding V; 3. Solving the system of algebraic equations and finding SC coefficients. 6. Test examples We consider here some test examples that will be numerically treated by the above proposed method. The numerical computations are carried out by the Mathematica xample: 6.1 Consider the following partial differential equation u, e x u1,0 = fx, y, x, y,, 6.1 to be the first test problem, with exact solution ux, y = Coshy Sech y T anh x where, the function fx, y takes the form fx, y = Sech x Coshy e x T anh x 1 + e x, 1 + Coshy and the subjected conditions are ux, y = 6 T anh x, at y, ux, y = 6 at x and at y, u0, 0 = 0, ux, 0 = at x. The fundamental matrix takes the form {Q 1,0 [ D x 1] + Q,1 [ D x D y 1]} A = F, we take m = n = 8, where, the approximate solution given by x, y = a 0,0 0,0x, y + a 0,1 0,1x, y + + a 8,8 8,8x, y.,

11 CMD Vol. 3, No. 3, 015, pp Then, after the augmented matrix of the system and conditions are computed, we obtain the coefficients solution as: a 0,0 = a 0,1 =... = a 0,8 = 0, a 1,0 = 0, a 1,1 = 0, a 1, = 1,..., a 1,8 = 0, a,0 = a,1 =... = a,8 = 0,. a 8,0 = a 8,1 =... = a 8,8 = 0, and the solution is given as: x, y = 1,x, y, x, y = e x 1 e x e y 1 e y +1 = 5 + 3Coshy Sech y which represent the exact solution of the problem. xample: 6. Consider the following differential equation [10] T anh x, u xy 1 + e x u 4e y y = 1 + e x 1 + e y, x, y,, 6. with conditions u y 0, y = 0, ux, 0 = 0. The fundamental matrix takes the form {Q 0,1 [ D y 1] + Q 1,1 [ D x 1 D y 1]} A = F, we take m = n = 4, where the approximate solution given by x, y = a 0,0 0,0x, y + a 0,1 0,1x, y + + a 4,4 4,4x, y, then, after the augmented matrix of the system and conditions are computed, we obtain the coefficients: a 0,0 = a 0,1 =... = a 0,4 = 0, a 1,0 = 0, a 1,1 = 1 4, a 1, = a 1,3 = a 1,4 = 0, a,0 = a,1 =... = a 4,4 = 0, and the solution is given by: x, y = 1 4 1,1x, y, or x, y = 1 [ e x 1 e y ] 1 4 e x + 1 e y = + 1 e x+y e x e y + 1 e x + 1 e y + 1,

12 158 M. RAMADAN, K. RASLAN, T. L DANAF, AND M. ABD L SALAM which represent the exact solution of q 6.. On the other hand, the approximate solution given in [10] at n = m = 15 the approximate solution doesn t give the exact solution. xample: 6.3 The Cauchy problem [16], for the one-dimensional homogeneous wave equation is given by u yy c u xx = 0, < x <, y [0,, u x, 0 = fx, u y x, 0 = gx, < x <. 6.3 The solution of this problem can be interpreted as the amplitude of a sound wave propagating in very long and narrow pipe, which in practice can be considered as one-dimensional infinite medium. The initial conditions f, g are given functions that represent the amplitude ux, y and the velocity u y of the string at time y = 0. The exact solution of 6.3 is given by DAlemberts formula ux, y = 1 [fx + cy + fx cy] + 1 c x+cy x cy gsds. Thus, if we take fx = Sechx and gx = 0, and applying our present method to solve 6.3, at n = m = 8, and 10 by using double SC collocation points, we obtain the approximate solution x, y. In Table 1, the exact and approximate solutions are listed according to different values of x, y. The calculation of L norm L = h I i=0 ui i presented in Table, shows that the grater n, m give us good accuracy at step size h = 0.1, x [, ], y [0, 1]. In Figure 1 we seek contour plots of the exact and approximate solutions n = m = 8, 10, such that x [, ], y [0, 1]. Also the error functions for n = m = 8 and 10 are plotted in Figure.

13 CMD Vol. 3, No. 3, 015, pp Figure 1. The contour plots of the exact and approximate solutions a contour plot of the exact solution b contour plot n = m = c contour plot n = m = Figure. The error function of exact and approximate solutions a rror functions for n = m = 8 b rror functions for n = m = 10

14 160 M. RAMADAN, K. RASLAN, T. L DANAF, AND M. ABD L SALAM Table. 1 comparing the approximate and exact solution x y xact Our method Abs error Our method Abs error solution n = m = 8 n = m = xample: 6.4 Table. Comparing the L -norm our method n = m = 8 n = m = 10 Let us consider the Poisson equation [, 16] u = f x, y, 0 x, y Poisson equation arises in steady state heat problems with time independent heat sources, where the Dirichlet boundary conditions in general form is u0, y = f 1 y, u1, y = f y, If we chose the exact solution to be as then, we find ux, 0 = g 1 x, ux, 1 = g x. ux, y = 1 + e x e y 1, f 1 y = ey 1, g 1 x = ex 1, f y = 1 + e y e 1, g x = 1 + e e x 1. Appling our present method to solve 6.4, at n = m = 8 by using double SC collocation points, we obtain the approximate solution

15 CMD Vol. 3, No. 3, 015, pp x, y = 0.50,0x, y 0.150,1x, y 0.151,0x, y ,1x, y. By simplifying the previous relation we reach to x, y = 1 + e x e y 1, which represent the exact solution of Poisson equation 6.4 with the connected conditions. 7. Conclusion In this paper, a collocation method for solving high-order linear partial differential equations with variable coefficients under more general form of conditions is investigated. The method is based on the approximation by truncated double exponential second kind Chebyshev SC series, and the definition of the partial derivatives is presented. All principles and properties of this type are derived and introduced by us as new definitions. The definition of the partial derivatives of SC functions is presented and derived as new operational matrices of derivatives. The PDs and conditions are transformed into block matrix equations, which correspond to system of linear algebraic equations with unknown SC coefficients, by using SC collocation points. The generalized invers is used to solve this linear system and to find the SC coefficients. Illustrative examples are used to demonstrate the applicability and the effectiveness of the proposed technique. In addition, an interesting feature of this method is to find the analytical exact solution if the equation has an exact solution of rational exponential form. The method can also be extended to high-order nonlinear partial differential equation with variable coefficients, but some modifications are required. Acknowledgment The authors sincerely thank the editors, reviewers and everyone who provide the advice, support, help and useful comments. The product of this research paper would not be possible without all of them. References [1] W. M. Abd-lhameed, Y. H. Youssri and. H. Doha, A novel operational matrix method based on shifted Legendre polynomials for solving second-order boundary value problems involving singular, singularly perturbed and Bratu-type equations, Math. Sci., 9 015,

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