[21] B. J. McCartin, Theory of exponential splines, Journal of Approximation Theory 66 (1991) 1 23.

Transcription

1 15 T. Ak, S. B. G. Karakoc, H. Triki, Numerical simulation for treatment of dispersive sallow water waves wit Rosenau- KdV equation, Te European Pysical Journal Plus 131 (1 ( M. Abbas, A. Abd. Majid, A. I. Md. Ismail, A. Rasid, Te application of cubic trigonometric B-spline to te numerical solution of te yperbolic problems, Appl. Mat. Comput. 39 ( M. Abbas, A. Abd. Majid, A. I. Md. Ismail, A. Rasid, Numerical metod using cubic trigonometric B-Spline tecnique for non-classical diffusion problem, Abstr. Appl. Anal. 14 (14, Article ID 84968, T. Nazir, M. Abbas, A. I. Md. Ismail, A. Abd. Majid, A. Rasid, Te numerical solution of advection diffusion problems using new cubic trigonometric B-splines approac, Applied Matematical Modelling 4 ( S. M. Zin, A. Abd. Majid, A. I. Md. Ismail, M. Abbas, Cubic Trigonometric B-spline Approac to Numerical Solution of Wave Equation, International Journal of Matematical, Computational, Pysical, Electrical and Computer Engineering 8 (1 (14. M. Yaseen, M. Abbas, A. I. Md. Ismail, T. Nazir, A cubic trigonometric B-spline collocation approac for te fractional sub-diffusion equations, Applied Matematics and Computation 93 ( B. J. McCartin, Teory of exponential splines, Journal of Approximation Teory 66 ( M. Sakai, R. A. Usmani, A class of simple exponential B-splines and teir application to numerical solution to singular perturbation problems, Numer. Mat. 55 ( S. C. S. Rao, M. Kumar, Exponential b- spline collocation metod for self-adjoint singularly perturbed boundary value problems, Appl. Numer. Mat. 58 (1 ( R. D. Multire, Solution exponential b- splines and singularly perturbed boundary problem, Numer. Algoritms 47 ( R. Moammadi, Exponential b-spline solution of convection-diffusion equations, Appl. Mat. 4 ( I. Dag, O. Ersoy, Exponential cubic b-spline algoritm for equal widt equation, Advanced Studies in Contemporary Matematics 5(4 ( O. Ersoy, I. Dag, Te exponential cubic b-spline algoritm for Korteweg-de vries equation, Advances in Numerical Analysis 15 (15, Article ID 36756, I. Dag, O. Ersoy, Numerical solution of generalized Burgers-Fiser equation by exponential cubic bspline collocation metod, AIP Conference Proceedings 1648 ( R. Moammadi, Exponential b-spline collocation metod for numerical solution of te generalized regularized long wave equation, Cin. Pys. B 4 (5 ( I. Dag, O. Ersoy, Te exponential cubic B-spline algoritm for Fiser equation, Caos, Solitons and Fractals 86 ( O. Ersoy, I. Dag, N. Adar, Te Exponential Cubic B-spline Algoritm for Burgers s Equation, arxiv: Xiv.org. 68

2 Journal of te Egyptian Matematical Society Volume (6 - Issue (1-18 ISSN:111-56X DOI:1.168/joems NUMERICAL TREATMENT OF FIRST ORDER MATRIX DIFFERENTIAL EQUATIONS USING DIFFERENT CUBIC B-SPLINE FUNCTIONS K. R. Raslan 1, A. R. Hadoud and M. A. Saalan 3 1 Faculty of Science, Al-Azar University, Cairo, Egypt Faculty of Science, Menoufia University, Sebein El-Koom, Egypt 3 Higer Tecnological Institute, Tent of Ramadan City, Egypt Received 18/3/18 Revised 16/3/18 Accepted 9/5/18 Abstract Te aim of te present paper is to present numerical treatments for solving Sylvester and Riccati matrix differential equations of first order wit polynomial, exponential and trigonometric cubic B-spline metods. Exactness and accuracy of te proposed metods are illustrated by calculating te maximum errors. Te results of numerical experiments sown by tese metods are convenient to be implemented and effective numerical tecnique for solving matrix differential equations. Keywords: Sylvester matrix differential equations, Riccati matrix differential equations, Polynomial cubic B-spline, Exponential cubic B-spline, Trigonometric cubic B-spline, Kronecker product, and Frobenius norm. Matematics Subject Classification. 41A15, 65D7, 65M7, 65M1. 1 Introduction Given a boundary matrix differential equation of first order } U (τ = f (τ, U(τ, a τ b, (1 U(a = U a, U(b = U b were matrices U a, U b, U(τ C m n and matrix function f : a, b C m n C m n, different examples of problem (?? can be found in 1. Sylvester matrix differential equations arise in many fields of science and engineering, 3, and Riccati matrix differential equations emerge a lot trougout science, applied matematics and engineering. In particular, tey play major roles in optimal control, filtering and estimation 4-6 and in solving linear, two-point boundary value problems of ordinary differential equations 7-1. If Y C m n andx C p q, we define te Kronecker product by Y X11 Y X = y 11 X y 1n X..... y m1 X y mn X Te column vector operator on a matrix Y C n m is given by 11: V ec(y = Y 1. Y m, were Y k =. ( Also, te derivative of a matrix U C m n wit respect to a matrix V C p q is defined by 11: v V = v 1q v. n 1n v n...., were =. v.... n v p1 v pq Y 1k. Y mk m1 v n. (3 mn v n. (4 Te derivative of a matrix product V C p q and U C q v wit respect to anoter matrix W C m n is given by 11: V U W = V W I n U + I m V W, (5 69

3 were te identity matrices of dimensions m and n denoted by I m and I n respectively. Te cain rule and derivative of a Kronecker product of matrices V U wit respect to a matrix W are given by 11: V ec (U T W V = (V U W V I m + I q W, (6 V ec (U = V W U + I m U 1 W V I n U, (7 were V C p q, U C u v, W C m n and U 1, U are permutation matrices. Te frobenius norm of U C m n is given by 1: m n U F = u ij. (8 Te Frobenius norm and -norm olds 1: i=1 j=1 U U F n U, (9 Cubic splines are discussed in 13-15, matrix differential equations are studied in and exponential cubic B-splines are piecewise polynomial functions containing a free parameter and its properties are presented in. Te exponential and trigonometric cubic B-spline metods are studied to solve numerical solutions of various ordinary and partial differential equations 1-7 and te sextic and septic B-spline metods are introduced to solve Rosenau-KdV equation 8, 9. Tis paper is organized as follows: In section, we present te polynomial, exponential and trigonometric cubic B-spline metods. In section 3, some numerical examples are discussed. Finally, te conclusion of tis study is given in section 4. Description of cubic B-spline metods Firstly, we assume tat te problem domain a, b is equally divided into N subintervals τ i, τ i+1, i =, 1,..., N 1 by te knots τ i = a + i were a = τ <τ 1 < <τ N 1 <τ N = b and te step size = b a N. Ten cubic B-spline collocation metods for solving matrix boundary value problems (1 numerically are presented..1 Polynomial cubic B-spline metod (PCBSM Te polynomial cubic B-spline can be defined as follows: B i (τ = 1 3 (τ τ i (τ τ i (τ τ i 1 3 (τ τ i (τ i+1 τ + 3 (τ i+1 τ 3 (τ i+1 τ 3 (τ i+ τ 3 (i = 1,,, N + 1. τ τ i, τ i 1, τ τ i 1, τ i, τ τ i, τ i+1, τ τ i+1, τ i+, elsewere. (1 We consider te spline function is interpolation to te solutions u (τ of te problem (1: N+1 u (τ = ω i (τ P CB i (τ ; 1 k n, 1 l m (11 i= 1 were constants ω i (τ s are be determined. To solve boundary matrix differential equation of first order, we find P CB i and P CB i at te nodal points are needed. Teir coefficients are summarized in Table 1. 7

4 Table 1. values of P CB i and P CB i. τ τ i τ i 1 τ i τ i+1 τ i+ P CB i P CB i 3 3 Using Eqs. (?? and (??, te values of u i and teir first derivatives at te knots are u i = ω i 1 +4 ω i + ω i+1, i =, 1,..., N. (1 u i = 3 ω i ω i+1 Substituting from Eq. (?? in Eq. (?? we find, 3 were i =, 1,..., N, k = 1,,..., n and l = 1,,..., m. and te boundary conditions are given as ω i ( ω i+1 = f i, ω i 1 +4 ω i + ω i+1, (13 ω 1 + 4ω + ω 1 = u a, ω N 1 +4 ω N + ω N+1 = u b. (14 Solving te system of Eqs. (?? in following matrix form ; ω 1 and ω N+1, te linear algebraic system of Eqs. (?? can be converted to te A ω = F, 1 k n, 1 l m. (15 were A is an (N + 1 (N + 1 matrix, ω is an N + 1 dimensional vector wit components ω i and te rigt and side F is an N + 1 dimensional vector; ω = ω, ω 1,..., ω T N, F = f, f 1,..., f N 1, f T N. (16. Exponential cubic B-spline metod (ECBSM Te exponential cubic B-spline can be defined as follows: w 1 (τ i τ 1 η (sin (η (τ i τ w + w 3 (τ i τ + w 4 e η(τi τ + w 5 e η(τi τ ECB i (τ = w + w 3 (τ τ i + w 4 e η(τ τi + w 5 e η(τ τi w 1 (τ τ i+ 1 η (sin (η (τ τ i+ τ τ i, τ i 1, τ τ i 1, τ i, τ τ i, τ i+1, τ τ i+1, τ i+, elsewere. (17 were, η ηc w 1 = (ηc S, w = ηc S, w 3 = η w 4 = 1 e η (1 C+S(e η 1 4 (ηc S(1 C, w 5 = 1 4 (i = 1,,, N + 1. C(C 1+S (ηc S(1 C, e η (C 1+S(e η 1 (ηc S(1 C, C = Cos (η, S = Sin (η, and η is a free parameter. We consider te spline function is interpolation to te solutions u (τ of te problem (??: N+1 u (τ = ζ i (τ ECB i (τ ; 1 k n, 1 l m, (18 i= 1 71

5 were constants ζ i (τ s are be determined. To solve boundary matrix differential equation of first order, we find ECB i and ECB i at te nodal points are needed. Teir coefficients are summarized in Table. Table. Values of ECB i and ECB i τ τ i τ i 1 τ i τ i+1 τ i+ ECB i β 1 1 β 1 ECB i β β were, β 1 = S η (ηc S, β = η (C 1 (ηc S, β 3 = η S (ηc S, β 4 = η S ηc S. Using Eqs. (17 and (18, te values of u i and teir first derivatives at te knots are u i = β 1 ζ i 1 + ζ i +β 1 ζ i+1, i =, 1,..., N. (19 u i = β ζ i 1 +β ζ i+1 Substituting from Eq. (?? in Eq. (?? we find, ( β ζ i 1 +β ζ i+1 = f i, β 1 ζ i 1 + ζ i +β 1 ζ i+1, ( were i =, 1,..., N, k = 1,,..., n and l = 1,,..., m. and te boundary conditions are given as 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. (1 Solving te system of Eqs. (1 in ζ 1 and following matrix form ; ζ N+1, te linear algebraic system of Eqs. (?? can be converted to te A ζ = F, 1 k n, 1 l m. ( were A is an (N + 1 (N + 1 matrix, ζ is an (N + 1 dimensional vector wit components ζ i and te rigt and side F is an (N + 1 dimensional vector; ζ = ζ, ζ 1,..., ζ T N, F = f, f 1,..., f N 1, f N T. (3.3 Trigonometric cubic B-spline metod (TCBSM Te trigonometric cubic B-spline can be defined as follows: ϕ 3 (τ i T CB i (τ = 1 ϕ (τ i ϕ (τ i ϑ (τ i + ϕ (τ i 1 ϑ (τ i+1 + ϕ (τ i 1 ϑ (τ i+ ϑ (τ i+ ϑ (τ i+ ϕ (τ i + ϑ (τ i+1 ϕ (τ i 1 + ϑ (τ i+1 ϕ (τ i ρ ϑ 3 (τ i+ τ τ i, τ i 1, τ τ i 1, τ i, τ τ i, τ i+1, τ τ i+1, τ i+, elsewere. (4 were, ρ = sin (i = 1,, 1,, N + 1. ( ( 3 sin ( sin, ϕ (τ i = sin ( τ τi, ϑ (τ i = sin ( τi τ. 7

6 we consider te spline function is interpolation to te solutions u (τ of te problem (1: N+1 u (τ = T i (τ B i (τ ; 1 k n, 1 l m, (5 i= 1 were constants T i (τ s are be determined. To solve boundary matrix differential equation of first order, we find T CB i and T CB i at te nodal points are needed. Teir coefficients are summarized in Table 3. Table 3. values of T CB i and T CB i. τ τ i τ i 1 τ i τ i+1 τ i+ T CB i Ω 1 Ω Ω 1 T CB i Ω 3 Ω 3 were, Ω 1 = sin ( sin( sin( 3, Ω = 1+cos(, Ω 3 = 3 Ω 5 = 3 cos ( sin ( (+4 cos(. 4 sin( 3, Ω 4 = 3(1+3 cos( 16 sin ( ( cos( +cos( 3, Using Eqs. (?? and (??, te values of u i and teir first derivatives at te knots are u i = Ω 1 T i 1 +Ω T i +Ω 1 T i+1, i =, 1,..., N. (6 u i = Ω 3 T i 1 +Ω 3 T i+1 Substituting from Eq. (?? in Eq. (?? we find, ( Ω 3 T i 1 +Ω 3 T i+1 = f i, Ω 1 T i 1 +Ω T i +Ω 1 T i+1, (7 were i =, 1,..., N, k = 1,,..., n and l = 1,,..., m. and te boundary conditions are given as Ω 1 Ω 1 T 1 +Ω T +Ω 1 T 1 = u a, T N 1 +Ω T N +Ω 1 T N+1 = u b. (8 Solving te system of Eqs. (?? in following matrix form; T 1 and T N+1, te linear algebraic system of Eqs. (?? can be converted to te A T = F, 1 k n, 1 l m. (9 were A is an (N + 1 (N + 1 matrix, T is an (N + 1 dimensional vector wit components T i and te rigt and side F is an (N + 1 dimensional vector; T = T, T T 1,..., T N, F = f, f 1,..., f N 1, f T N. (3 3 Numerical examples In tis section, we present examples of matrix differential equations of first order to explore te efficiency and accuracy of te proposed metods using Frobenius norm of te difference between approximate solution and exact solution at eac point in te interval a, b taking a suitable step size and te results are generated wit Matematica using Find Root function to solve te emerging algebraic equations. 73

7 Example 1. We examine te linear Sylvester matrix differential equation of te type 18 were, U (τ = A (τ U(τ + U(τB(τ + C(τ; a τ b, U(τ, A(τ, B(τ, C(τ C n n, (31 ( τe τ A (τ = τ Tis example as an exact solutionu (τ = ( τ, B (τ = ( e τ ( ( 1 + τ e, C (τ = τ τe τ 1 τe τ τ, τ 1.. Tus, we can compare our numerical estimates wit tis solution τ 1 to obtain te exact errors of te approximation wic summarized in Table 4 and figure 1. For a free equilibrium pointsu j, (j = 1,,..., 4, we find tat te Jacobian matrix of te Eq. (?? is and its eigenvalues are evaluated from te equation τe τ τ τe τ τ τ τ λ 4 τ e τ λ + τ 4 e τ =, (3 were λ 1, = τe τ/ and λ 3, 4 = τe τ/ ave distinct signs for any value of τ in te interval (, 1, ten te equilibrium points u j = of te Eq. (?? are unstable.; (j = 1,,..., 4. Table 4. Comparison of maximum absolute errors for Example 1. τ Ploynomial Cubic Exponential Cubic Trigonometric Cubic Cubic spline B-spline errors B-spline errors B-spline errors errors 18 (PCBSM (ECBSM (TCBSM (CSM Figure 1. Comparison of maximum absolute errors for Sylvester matrix differential equation in te interval, 1 wit step size =.1. 74

8 Example. We investigate te rectangular non-symmetric Riccati matrix differential equation of te type 18 U (τ = C(τ D(τU(τ U(τA(τ U(τB(τU(τ; U (τ C n m, U (τ C n m, A (τ C m m, B (τ C m n, C (τ C n m, D (τ C n n, τ a (33 were, ( ( ( τ τ 1 τ A (τ =, B (τ =, D (τ =, ( τ ( τ 1 τ τ τ e C (τ = τ + τe τ τ 3 τ ( e τ τ τ (1 τ ( + τ + τ 1 + τ (3 τ + e ( τ τ τ 4, τ.1. ( e τ Tis example as an exact solution U (τ = τ. Tus, we can compare our numerical estimates wit tis τ solution to obtain te exact errors of te approximation wic summarized in Table 5 and figure. For a free equilibrium points u j, (j = 1,,..., 4, we find tat te Jacobian matrix of te Eq. (33 is 1 + τ τ τ 1 τ τ τ τ, (34 τ τ and its eigenvalues are evaluated from te equation λ 4 + ( 1 + τ λ 3 + ( 1 4τ τ + τ 3 λ + ( τ τ 3 + τ 4 λ + ( + τ + τ τ 4 = ( were λ 1, = 1 1 3τ ± 1 + τ + τ 4τ 3 ( and λ 3, 4 = τ ± 1 + τ + τ 4τ 3 ave distinct sign for any value of τ in te interval (,.1, ten te equilibrium points u j = of te Eq. (33 are unstable.; (j = 1,,..., 4. Table 5. Comparison of maximum absolute errors for Example. τ Ploynomial Cubic Exponential Cubic Trigonometric Cubic Cubic spline B-spline errors B-spline errors B-spline errors errors 18 (PCBSM (ECBSM (TCBSM (CSM Conclusion In tis article, we ave examined sceme treat numerically wit te first-order matrix differential equations by cubic B- spline metod and exponential and trigonometric cubic B-spline metods. From te computational results, we can view tat te cubic B-spline and exponential and trigonometric cubic B-splines are summarized and easy to apply and te errors are acceptable. Te numerical experiments are compared wit te analytic solutions by finding Frobenius norm and are compared wit Ref. 18 as sown in Tables (3-5 and Figures (1,. Acknowledgments Te autors would like to express teir sincere tanks to te reviewers for teir careful reading, additions valuable scientific comments and suggestions. 75

9 Figure. Comparison of maximum absolute errors for Riccati matrix differential equation in te interval, 1 wit step size =.1. References 1 U. M. Ascer, R. M. M. Matteij, R. D. Russell, Numerical Solutions of Boundary Value Problems for Ordinary Differential Equations, Prentice Hall, New Jersey, L. Jodar, E. Ponsoda, Computing continuous numerical solutions of matrix differential equations, Comput. Mat. Appl. 9 (4 ( L. V. Fausett, Sylvester matrix differential equations: analytical and numerical solutions, International Journal of Pure and Applied Matematics, Volume 53 (1 ( H. Kwakernaak, R. Sivan, Linear Optimal Control Systems, Wiley Interscience, New York, J. L. Casti, Dynamical Systems and Teir Applications: Linear Teory, Academic Press, New York, L. Jodar, J. C. C. Lopez, Rational matrix approximation wit a priori error bounds for non-symmetric matrix Riccati equation wit analytic coefficients, IMA J. Numer. Anal. 18 (4 ( U. M. Ascer, R. M. Matteij, R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, I. Babuška, V. Majer, Te factorization metod for te numerical solution of two point boundary value problems for linear ODE s, SIAM J. Numer. Anal. 4 ( L. Dieci, M. R. Osborne, R. D. Russell, A Riccati transformation metod for solving linear BVPs. I: teoretical aspects, SIAM J. Numer. Anal. 5 ( L. Dieci, A Riccati transformation metod for solving linear BVPs. II: computational aspects, SIAM J. Numer. Anal. 5 ( A. Graam, Kronecker products and matrix calculus wit applications, Jon Wiley, New York, G. H. Golub and C. F. Van Loan, Matrix computations, second ed., Te Jons Hopkins University Press, Baltimore, MD, USA, F. R. Loscalzo and T. D. Talbot, Spline function approximations for solutions of ordinary differential equations, SIAM J. Numer. Anal. 4 (3 ( M. K. Kadalbajoo and K. C. Patidar, Numerical solution of singularly perturbed two-point boundary value problems by spline in tension, Appl. Mat. Comput. 131 ( E. A. Al-Said and M. A. Noor, Cubic splines metod for a system of tird-order boundary value problems, Appl. Mat. Comput. 14 (

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