A Numerical Iterative Method for Solving Systems of First-Order Periodic Boundary Value Problems

Abstract

The objective of this paper is to present a numerical iterative method for solving systems of first-order ordinary differential equations subject to periodic boundary conditions. This iterative technique is based on the use of the reproducing kernel Hilbert space method in which every function satisfies the periodic boundary conditions. The present method is accurate, needs less effort to achieve the results, and is especially developed for nonlinear case. Furthermore, the present method enables us to approximate the solutions and their derivatives at every point of the range of integration. Indeed, three numerical examples are provided to illustrate the effectiveness of the present method. Results obtained show that the numerical scheme is very effective and convenient for solving systems of first-order ordinary differential equations with periodic boundary conditions.

Analytical treatment for solving system of fuzzy IVPs using residual power series approach

“…Solutions of the FDEs can often be expressed in terms of series expansions. However, the RPS technique is an analytical as well as numerical method for solving different types of ordinary and partial differential equations, integral equation and integro-differential equation [7][8][9][10][11][12][13][14][15][16]. The methodology is effective and easy to construct power series solution for strongly linear and nonlinear systems of FIVPs without linearization, perturbation, or discretization [17][18][19][20][21][22][23].…”

“…Anyhow, when = 10 is used throughout the computations; the following are the first fifth terms of RPS approximation of Eqs. (16) and 17…”

“…Then we have the following system of ODE: 1 ′ ( ) = 0.05 − 0.5( 2 − 1200), 2 ′ ( ) = 0.05 − 0.5( 1 − 1200), 1 ′ ( ) = 0.5( 1 − 25), 2 ′ ( ) = 0.5( 2 − 25), (16) subject to the initial conditions 1 (0) = 20 + 5 , 2 (0) = 30 − 5 , 1 (0) = 550 + 50 , 2 (0) = 650 − 50 . (17)Using RPS method, taking 1,0 = 20 + 5 , 2,0 = 30 − 5 , 3,0 = 550 + 50 , and 4,0 = 650 − 50 as initial guess approximation, the Taylor series expansions of solutions for Eqs (16). and(17) are as follows…”

Fitted Reproducing Kernel Method for Solving a Class of Third-Order Periodic Boundary Value Problems

“…However, Al-Smadi et al (2014) have developed an iterative method for handling system of first-order PBVPs based on the RKHM. While Hopkins and Kosmatov (2007) have provided the existence of at least one positive solutions of PBVP in the form u′′′ (x) = f (x, u (x), u′(x), u′′ (x)).…”

Analytical-numerical method for solving a class of two-point boundary value problems

“…In fact, these equations, which have attracted considerable attention over the last two decades, are usually difficult to solve analytically, so it is required to obtain an efficient analytical-numerical solution. Therefore, many techniques arose in the studies existence and constructive approximation of solutions of such problems, Especially, those techniques that based on the reproducing kernel Hilbert space theory, for instance, the first-order Fredholm-Volterra IDEs [21,30], the second-order IDEs Fredholm or Volterra or of mixed type [16,19], and fourth-order IDEs [22,23]. On the other hand, the numerical solvability of other version of differential problems can be found in [8-13, 19, 20, 24-26].…”