A Numerical Method Based on RKHS for a Hyperbolic Initial-Boundary-Value Problem

Subject Areas : International Journal of Mathematical Modelling & Computations

Ebrahim Amini ^{1 *}

1 - Department of Mathematics, Payme Noor University, P. O. Box 19395-4697, Tehran,IRAN

Received: 2023-02-24 Accepted : 2023-09-16 Published : 2023-09-30

Keywords: Hyperbolic differential-integral equations, Initial and boundary conditions, Approximate solu-tion, Convergence analysis, Reproducing kernel Hilbert space,

Abstract :

The main aim of this article is to propose a computational method on the basis of the reproducing kernel Hilbert space method for solving a hyperbolic initial-boundary-value problem. The solution in reproducing kernel Hilbert space is constructed with series form, and the approximate solution vm is given as an m-term summation. Furthermore, convergence of the proposed method is presented which provides the theoretical basis of the proposed method. Finally, some numerical experiments are considered to demonstrate the efficiency and applicability of proposed method.

References:

[1] S. Zaremba, Sur le calcul numerique des founctions demandness dans le problems de dirichlet et le
problems hydrodynamique, Bull. Int. Acad. Sci. Cracovie (1908), 125-195.
REFERENCES 13
[2] J. Mercer, Functions of positive and negative type and their connection with the theory of integral
equations, Philos. Trans. R. Soc. Lond. 209 (1909), 415-446.
[3] N. Aronszajn, Theory of reproducing kernels, Harvard University, Cambridge, MA, (1951).
[4] M. Cui and Z. H. Deng, Numerical functional method in reproducing kernel space, The Publication
of Harbin Institute of Technology, China, (1988).
[5] M. Cui and Y. Yan, The representation of the solution of a kind of operator equation Au = f, Numer.
Math. J. Chinese Univ., 1 (1995), 82-86.
[6] M. Fardi, R. K. Ghaziani, and M. Ghasemi, The reproducing kernel method for some variational
problems depending on indefinite integrals, Math. Model. Anal. 21, (2016), 412-429.
[7] M. Ghasemi, M. Fardi, and R. K. Ghaziani, Numerical solution of nonlinear delay differential equations
of fractional order in reproducing kernel Hilbert space, Appl. Math. Comput. 268, (2015), 815-831.
[8] M. Fardi and M. Ghasemi, Solving nonlocal initial-boundary value problems for parabolic and hyperbolic integro-differential equations in reproducing kernel hilbert space, Numer. Methods Partial
Differ. Equ. 33, (2016), 174-198.
[9] S. Vahdati, M. Fardi, and M. Ghasemi, Option pricing using a computational method based on
reproducing kernel, J. Comput. Appl. Math. 328, (2018), 252-266.
[10] M. Xu, L. Zhang, and E. Tohidi, A fourth-order least-squares based reproducing kernel method for
one-dimensional elliptic interface problems, Appl. Numer. Math. 162, (2021), 124-136.
[11] M. Fardi, I. Pishkar, and J. Alidousti, Numerical investigation of the MHD suction-injection model
of viscous fluid using a kernel-based method, Arch. Appl. Mech. 91, (2021), 4205-4221.
[12] M. Fardi and Y. Khan, Numerical simulation of squeezing Cu-water nanofluid flow by a kernel-based
method, Int. J. Model. Simul. Sci. Comput. 13(1), (2022), 2250005.
[13] E. Amini, Optimal Control Problems: Convergence and Error Analysis in Reproducing Kernel Hilbert
Spaces, Control and Optimization in Applied Mathematics, 6(2), ( 2021), 53-77.
[14] E. Amini, A. Taheri, & M. Abedini, Using reproducing kernal for solving the Lane-Emden equations,
arising Astrophysics, International Journal of Mathematical Modelling & Computaions, 10(3), (2020),
179-202.
[15] M. Cui , and Y. Lin , Nonlinear numerical analysis in the reproducing Kernel space, NY: Nova Science
Publ, New York, (2009).
[16] M. Fardi, and M. Ghasemi, Numerical solution of singularly perturbed 2D parabolic initial-boundaryvalue problems based on reproducing kernel theory: Error and stability analysis, Numerical Methods
for Partial Differential Equations, 38(4), (2020), 876-903.
[17] F. Geng, and M. Cui, Solving singular nonlinear second-order periodic boundary value problems in
the reproducing kernel space, Applied Mathematics and Computation, 192(2), (2007), 389-398.
[18] R. D. Russell and L. F. Shampine, A collocation method for boundary value problems, Numer. Math.,
19(1), (1972), 1-28.
[19] Y. Zhou, M. Cui, Y. Lin, Numerical algorithm for parabolic problems with non-classical conditions,
J. Comput. Appl. Math., 230(2), (2009), 770-780