Numerical Solution and Error Analysis for Linear and Nonlinear Delay Differential Equations
Subject Areas : Statistics
Ebrahim Amini
1
,
Ali Ebadian
2
1 - Department of Mathematics, Payme Noor University,P. O. Box 19395-4697 Tehran, Iran
2 - Department of Mathematis, Urmia University, Urmia, Iran
Keywords: آنالیز همگرایی, هستهی بازتولید, معادلهی دیفرانسیل تاخیری, آنالیز خطا,
Abstract :
In this paper, we obtain the solution of linear and nonlinear delay differential equations in reproducing kernel space. For this purpose, regarding the equation and conditions governing it, a linear operator is defined and subsequently an orthonormal complete system for reproducing kernel space is obtained by using the adjoint operator and reproducing kernel function. Then, the solution of these equations is obtained in the form of a series of the basic functions. Indeed, the analytical solution is represented by infinite series, and the approximate solution is obtained by using an iterative method. As one of the main aims, the convergence analysis and error behavior are discussed for the proposed method. Finally, some numerical examples are studied to demonstrate the validity and applicability of the proposed method. The obtained results of the proposed method are compared with the exact solutions and the earlier works. The outcomes from numerical examples illustrate that the proposed method is very effective and convenient.
[1] Aronszajn, N. (1951). Theory of reproducing kernels. Cambridge, MA: Harvard niversity.
[2] Cui, M. and Geng, F. (2007). A computational method for solving one-dimensional variable-coefficient Burgers equation. Applied Mathematics and Computation, 188, 1389-1401.
[3] Cui, M. and Lin, Y. (2009). Nonlinear numerical analysis in the reproducing Kernel space, NY: Nova Science Publ, New York, .
[4] Doha, E.H., Bhrawy, A.H., Blaleanu, D. and Hafez, R. M. (2014). A new Jacobi rational–Gauss collocation method for numerical solution of generalized pantograph equations. Applied Numerical Mathematics, 77, 43-54.
[5] Fardi, M. Ghasemi, M. (2016). Solving nonlocal initial-boundary value problems for parabolic and hyperbolic integro-differential equations in reproducing kernel hilbert space, Numerical Methods for Partial Differential Equations, 3, 174-198.
[6] Fardi, M., Ghasemi, M. and Khoshsiar Ghaziani, R. (2016). The Reproducing Kernel Method for Some Variational Problems Depending on Indefinite Integrals, Mathematical Modelling and Analysis, 21, 412-429.
[7] Geng, F. and Cui, M. (2012). A reproducing kernel method for solving nonlocal fractional boundary value problems, Applied mathematics Letters, 25, 818-823.
[8] Geng, F. Z. and Li, X. M. (2012). A New Method for Riccati Differential Equations Based on Reproducing Kernel and Quasiline-arization Methods, Abstract and Applied Analysis, 1-8.
[9] Ghasemi, M., Fardi, M. and Khoshsiar Ghaziani, R. (2015). Numerical solution of nonlinear delay differential equations of fractional order in reproducing kernel Hilbert space, Applied Mathematics and Computation, 268, 815-831.
[10] Ismail, H. N., Raslan, K. and Rabboh, A. A. (2004). Adomian decomposition method for Burgers–Huxley and Burgers–Fisher equations. Applied Mathematics and Computation, 159, 291-301.
[11] Jiang, W., Cui, M. and Lin, Y. (2010). Anti-periodic solutions for Rayleigh-type equations via the reproducing kernel Hilbert space method, Communications in Nonlinear Science and Numerical Simulation, 15, 1754-1758.
[12] Tohidi. E., Bhrawy, A. H. and Erfani, K. (2013). A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation, Applied Mathematical Modelling, 37, 4283-4294.
[13] Vahdati, S. Fardi, M. and Ghasemi, M. (2018). Option pricing using a computational method based on reproducing kernel, Journal of Computational and Applied Mathematics, 328, 252-266.
[14] Yalinbas, S., Aynigul, M. and Sezer, M. (2011). A collocation method using Hermi- -te polynomials for approximate solution of pantograph equations. Journal of the ranklin Institute, 348,1128-1139.
[15] Zhou, Y., Cui, M. and Lin, Y. (2009). Numerical algorithm for parabolic problems with non- Classical conditions, Journal of Computational and Applied Mathematics, 230, 770–780.