The exponential and trigonometric cubic B-spline methods for second order matrix differential equations

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1 1 JACM 3, No. 1, 1-1 (218) Journal of Abstract and Computational Mathematics The exponential and trigonometric cubic B-spline methods for second order matrix differential equations K. R. Raslan 1 A. R. Hadhoud 2 and M. A. Shaalan 3 1 Faculty of science, Al-Azhar University, Cairo, Egypt 2 Faculty of Science, Menoufia University, Shebein El-Koom, Egypt 3 Higher Technological Institute, Tenth of Ramadan City, Egypt Received: 28 Jan 218, Accepted: 22 Feb 218 Published online: 7 May 218 Abstract: The goal of the present paper is to present numerical treatments for solving matrix differential equations of second order using exponential and trigonometric cubic B-splines. Efficiency and accuracy of the proposed methods are illustrated by calculating the maximum errors. The results of numerical experiments shown by these methods are convenient to be implemented and effective numerical technique for solving matrix differential equations. Keywords: Matrix differential equations, exponential cubic B-spline, trigonometric cubic B-spline, kronecker product, frobenius norm. 1 Introduction Given matrix boundary value problems } U (t) = f (t,u(t),u (t)),a t b,[a,b R, (1) U(a) = U a,u(b) = U b where matrices U a,u b,u(t) C m n and matrix function f : [a,b C m n C m n C m n, are recurrent in various phenomena in physics and engineering. Equation (1) is considered as the statement of Newton s law of motion for coupled mechanical system. Usually models of this kind recurrently appear in molecular dynamics, quantum mechanics and for scattering methods, where one solves scalar or vectorial problems subject to boundary value conditions [1,6. We define the Kronecker product of Y C m n andx C p q, denoted byy X [7 y 11 X y 1n X Y X =....., (2) y m1 X y mn X Corresponding author abozeid87@yahoo.com c 218 BISKA Bilisim Technology

2 JACM 3, No. 1, 1-1 (218) / 2 The column vector operator on a matrix Y C n m is given by [7. Vec(Y ) =. Y 1 Y m wherey k = Also, the derivative of a matrix U C m n with respect to a matrix V C p q is defined by [7.. Y 1k Y mk. (3) U V = U v 11. U v p1 U v 1q.... U v pq U,where = v n U 11 v n. U m1 v n U 1n v n..... (4) U mn v n The derivative of a matrix product V C p q and U C q v with respect to another matrix W C m n is given by [7. VU W = V W [I n U + [I m V U W, (5) where the identity matrices of dimensions m and n denoted by I m and I n respectively. The chain rule and derivative of a Kronecker product of matrices V Uwith respect to a matrix W are given by [7. [ W V = [Vec(U) T [ W I m + I q, (6) V [Vec(U) (V U) W = V [ U W U + [I m U 1 W V [I n U 2, (7) where V C p q, U C u v, W C m n and U 1, U 2 are permutation matrices. The frobenius norm of U C m n is given by [8. U F = m i=1 n j=1 The following relationship between the 2-norm and Frobenius norm holds [8. ui j 2. (8) U 2 U F n U 2. (9) Cubic splines are discussed in [9,1, matrix differential equations are studied in [11,14 and exponential cubic B-splines are piecewise polynomial functions containing a free parameter and its properties are presented in [15. The exponential and trigonometric cubic B-spline methods and spline with different basis as sextic are studied to solve numerical solutions of various ordinary and partial differential equations [16,22. This paper is organized as follows; In section 2, we present the exponential and trigonometric cubic B-spline methods. In section 3, some numerical examples are discussed. Finally, the conclusion of this study is given in section 4. 2 Description of cubic B-spline methods Firstly, we assume that the problem domain [a,b is equally divided into N subintervals [t i,t i+1, i =, 1,..., N 1 by the knots t i = a + ih where a = t <t 1 < <t N 1 <t N = b and the step size h = b a N. c 218 BISKA Bilisim Technology

3 3 K. R. Raslan: The exponential and trigonometric cubic B-spline methods Exponential cubic B-spline method (ECBSM) The exponential cubic B-spline can be defined as follows w 1 [(t i 2 t) η 1 (sinh(η (t i 2 t))) w 2 + w 3 (t i t) + w 4 e η(ti t) + w 5 e η(t i t) ECB i (t) = w 2 + w 3 (t t i ) + w 4 e η(t ti) + w 5 e η(t t i) w 1 [(t t i+2 ) η 1 (sinh(η (t t i+2))) (i = 1,,,N + 1), t [t i 2,t i 1, t [t i 1,t i, t [t i,t i+1, t [t i+1,t i+2, elsewhere. (1), where η ηhc w 1 = 2(ηhC S),w 2 = ηhc S,w 3 = η [ 2 w 4 = 1 4,w 5 = 1 4 e ηh (1 C)+S(e ηh 1) (ηhc S)(1 C) [ C(C 1)+S 2 (ηhc S)(1 C) [, e ηh (C 1)+S(e ηh 1) (ηhc S)(1 C),C = Cosh(ηh),S = Sinh(ηh), and η is a free parameter. We consider the spline function as interpolation to the solutions u (t) of the problem (1). u (t) = N+1 i= 1 ζ i (t)ecb i (t);1 k n, 1 l m, (11) where constants ζ i (t) s are be determined. To solve matrix boundary value problems of second order, we find ECB i,ecb i andecb i at the nodal points are needed. Their coefficients are summarized in Table 1. Table 1: values of ECB i,ecb i and ECB i t i 2 t i 1 t i t i+1 t i+2 ECB i β 1 1 β 1 ECB i β 2 β 2 ECB i β 3 β 4 β 3 where β 1 = S ηh 2(ηhC S), β 2 = η (C 1) 2(ηhC S), β 3 = η 2 S 2(ηhC S), β 4 = η2 S ηhc S. Using equations (1) - (11), the values of u i and their derivatives up to second order at the knots are Substituting from (12) in (1), we find β 3 ζ i 1 +β 4 u i = β 1 ζ i 1 + ζ i +β 1 ζ i+1 ζ i+1 ζ i+1 u i = β 2 ζ i 1 +β 2 u i = β 3 ζ i 1 +β 4 ζ i +β 3,i =,1,...,N. (12) ( ζ i +β 3 ζ i+1 = f ih,β 1 ζ i 1 + ζ i +β 1 ζ i+1, β 2 ζ i 1 +β 2 ζ i+1 ), c 218 BISKA Bilisim Technology

4 JACM 3, No. 1, 1-1 (218) / 4 and the boundary conditions are given as i =,1,...,N, k = 1,2,...,nandl = 1,2,...,m, (13) β 1 ζ 1 + β 1 ζ +β 1 ζ 1 = u a, ζ N 1 + ζ N +β 1 ζ N+1 = u b. (14) Solving the system of equations (14) in ζ 1 and ζ N+1, the linear algebraic system of equations (13) can be converted to the following matrix form Aζ = F,1 k n, 1 l m, (15) where A is an (N + 1) (N + 1) matrix, ζ is an (N + 1) dimensional vector with components ζ i and the right hand side F is an (N + 1) dimensional vector. [ ζ = ζ, ζ 1,..., ζ T [ N, F = f, f 1,..., f N 1, f T N. (16) 2.2 Trigonometric cubic B-spline method (TCBSM) The trigonometric cubic B-spline can be defined as follows ϕ 3 (t i 2 ) TCB i (t) = 1 ϕ (t i 2 )[ϕ (t i 2 )ϑ (t i ) + ϕ (t i 1 )ϑ (t i+1 ) + ϕ 2 (t i 1 )ϑ (t i+2 ) ϑ (t ρ i+2 )[ϑ (t i+2 )ϕ (t i ) + ϑ (t i+1 )ϕ (t i 1 ) + ϑ 2 (t i+1 )ϕ (t i 2 ) ϑ 3 (t i+2 ) (i = 1,,1,,N + 1), t [t i 2,t i 1, t [t i 1,t i, t [t i,t i+1, t [t i+1,t i+2, elsewhere. (17) where ρ = sin ( ) ( ) h 3h sin(h)sin, ϕ (t i ) = sin 2 2 ( t ti 2 ), ϑ (t i ) = sin ( ti t 2 ). We consider the spline function as interpolation to the solutions u (t) of the problem (1) u (t) = N+1 i= 1 τ i (t)b i (t) ;1 k n, 1 l m, (18) where constants τ i (x) s are be determined. To solve matrix boundary value problems of second order, we find TCB i,tcb i andtcb i at the nodal points are needed. Their coefficients are summarized in Table 2. c 218 BISKA Bilisim Technology

5 5 K. R. Raslan: The exponential and trigonometric cubic B-spline methods... Table 2: values of TCB i,tcb i andtcb i t i 2 t i 1 t i t i+1 t i+2 TCB i Ω 1 Ω 2 Ω 1 TCB i Ω 3 Ω 3 TCB i Ω 4 Ω 5 Ω 4 where Ω 1 = sin2 ( 2) h sin(h)sin( 3h 2 ), Ω 2 2 = 1+cos(h), Ω 3 3 = 3cos Ω 5 = 2 ( h 2) sin 2 (. h 2)(2+4cos(h)) 4sin( 3h 2 ), Ω 4 = 3(1+3cos(h)) 16sin 2 ( h 2)(2cos( h 2)+cos( 3h 2 )), Using equations (17) - (18), the values of u i and their derivatives up to second order at the knots are Substituting from (19) in (1) we find Ω 4 τ i 1 +Ω 5 u i = Ω 1 τ i 1 +Ω 2 τ i +Ω 1 u i = Ω 3 τ i 1 +Ω 3 u τ i+1 i = Ω 4 τ i 1 +Ω 5 τ i +Ω 4 τ i+1 τ i+1,i =,1,...,N. (19) ( τ i +Ω 4 τ i+1 = f ih,ω 1 τ i 1 +Ω 2 τ i +Ω 1 τ i+1, Ω 3 τ i 1 +Ω 3 τ i+1 ), and the boundary conditions are given as i =,1,...,N, k = 1,2,...,nandl = 1,2,...,m, (2) Ω 1 τ 1 +Ω 2 Ω 1 τ +Ω 1 τ 1 = u a, τ N 1 +Ω 2 τ N +Ω 1 τ N+1 = u b. (21) Solving the system of equations (21) in τ 1 and τ N+1, the linear algebraic system of equations (2) can be converted to the following matrix form A τ = F,1 k n, 1 l m, (22) where A is an (N + 1) (N + 1) matrix, τ is an (N + 1) dimensional vector with components τ i and the right hand side F is an (N + 1) dimensional vector, [ T [ τ = τ, τ 1,..., τ N, F = f, f 1,..., f N 1, f T N. (23) 3 Numerical examples In this section, three examples of matrix differential equations of second order are presented to show efficiency and accuracy of the proposed methods using Frobenius norm of the difference between approximate solution and exact solution at each point in the interval [,1 taking h =.1and the results are generated with Mathematical using Find Root function to solve the emerging algebraic equations. c 218 BISKA Bilisim Technology

6 JACM 3, No. 1, 1-1 (218) / 6 Example 1. We examine non-linear differential vector system [13. u 1 (t) = 1 cos(t) + sin(u 2 (t)) + cos(u 2 (t)) u 2 (t) = u 1 (t) 2 5 sin 2 (t) u 1 () = 1, u 1 (1) = cos(1), u 2 () =, u 2 (1) = π t 1. (24) ( ) cos(t) This example has an exact solution U (t) =. Thus, we can contrast our numerical estimates with the exact πt solution to get the exact errors of the approximation which briefed in Table 3 and figure 1. For free equilibrium points u j,( j = 1,2,...,4), we find that the Jacobian matrix of (24) , (25) and its eigenvalues are evaluated from the equation λ 4 = where λ j =, then the equilibrium points u j = of (24) are unstable ( j = 1,2,...,4). Table 3: Comparison of maximum absolute errors for Example 1. t Exponential Cubic B- (ECBSM) Trigonometric Cubic B- (TCBSM) Cubic (CSM) [ c 218 BISKA Bilisim Technology

7 7 K. R. Raslan: The exponential and trigonometric cubic B-spline methods... Fig. 1: Comparison of maximum absolute errors for non-linear differential vector system in the interval [, 1 with step size h =.1. Example 2. We investigate incomplete differential system of second order [13. U (t) + AU (t) =, t 1. (26) ( ) ( ) 1 sin(t) Where A = and corresponding exact solution U (t) =. Thus, we can observe the exact errors 2 1 t cos(t) sin(t) of the approximation which briefed in Table 4 and figure 2. For the only equilibrium point u j =,( j = 1,2,...,8), we find that the Jacobian matrix of (26) is (27) and its eigenvalues are evaluated from the equation λ 8 + 4λ 6 + 6λ 4 + 4λ =, where λ j = ±i, then (26) has a unique equilibrium point u j = is an stable ( j = 1,2,...,8). Example 3. We consider the following problem of second order polynomial matrix equation [13. U (t) + A U (t) + A 1 U (t) =, t 1. (28) c 218 BISKA Bilisim Technology

8 JACM 3, No. 1, 1-1 (218) / 8 Table 4: Comparison of maximum absolute errors for Example 2. t Exponential Cubic B- (ECBSM) Trigonometric Cubic B- (TCBSM) Cubic spline errors (CSM) [ Fig. 2: Comparison of maximum absolute errors for incomplete differential system of second order in the interval [, 1 with step size h =.1. ( ) ( ) ( ) 1 1 e t 1 + e t te t Where A =,A 1 = and the exact solution U (t) = 2 1 e t. Thus, we are illustrated the exact errors at each point in Table 5 and figure 3. For free equilibrium points u j,( j = 1,2,...,8), we find that the Jacobian matrix of (28) is (29) c 218 BISKA Bilisim Technology

9 9 K. R. Raslan: The exponential and trigonometric cubic B-spline methods... and its eigenvalues are evaluated from the equation λ 8 6λ λ 6 2λ λ 4 6λ 3 + λ 2 = where λ 1,2 = and λ j = 1,( j = 3,...,8), then the equilibrium points u j of (28) are unstable.; ( j = 1,2,...,8). Table 5: Comparison of maximum absolute errors for Example 3. t Exponential Cubic B- (ECBSM) Trigonometric Cubic B- (TCBSM) Cubic spline errors (CSM) [ Fig. 3: Comparison of maximum absolute errors for second order polynomial matrix equation in the interval [, 1 with step size h =.1. 4 Conclusion In this article, we have examined scheme treat numerically with the second-order matrix differential equations by exponential and trigonometric cubic B-splines. From the computational results, we can see that the exponential and c 218 BISKA Bilisim Technology

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