Solving Systems of Volterra Integral and Integrodifferential Equations with Proportional Delays by Differential Transformation Method

Abstract

In this paper, the differential transformation method is applied to the system of Volterra integral and integrodifferential equations with proportional delays. The method is useful for both linear and nonlinear equations. By using this method, the solutions are obtained in series forms. If the solutions of the problem can be expanded to Taylor series, then the method gives opportunity to determine the coefficients of Taylor series. Hence, the exact solution can be obtained in Taylor series form. In illustrative examples, the method is applied to a few types of systems.

Solutions of system of Volterra integro‐differential equations using optimal homotopy asymptotic method

“…Using the values of these constants in Equations 26 and 27, the approximate solution becomes u s ð Þ = 0:056747564775805864 + 0:9432524352241941e s + s − s 3…”

“…The exact solutions of these systems are difficult to find. In finding the approximate solutions, different approaches in the literature have been used such as homotopy perturbation method (HPM), 1 homotopy analysis method (HAM), 2 differential transformation method (DTM), 3 Genocchi polynomials (GP), 4 reproducing kernel Hilbert space method (RKHSM), 5 Chebyshev wavelets method (CWM), 6 fixed point method (FPM), 7 Sinc-collocation method (SCM), 8 Haar functions method (HFM), 9 Adomian decomposition method (ADM), 10 and relaxed Monte Carlo method (RMCM). 11 In the present attempt, optimal homotopy asymptotic method (OHAM) is used, which was introduced by Marinca et al 12,13 for the solution of differential equations.…”

Hybrid functions for numerical solution of fractional Fredholm‐Volterra functional integro‐differential equations with proportional delays

“…Consider the linear system of fractional Volterra functional integro‐differential equation with proportional delays with the initial conditions u 1 (0) = 1 and u 2 (0) = 0. The exact solution of this problem for ν = γ = 1 is u 1 ( t ) = cos ( t ), u 2 ( t ) = sin ( t ). Table shows the absolute errors between the exact and approximate solutions for various values of M . Figure A,B demonstrates the approximation of u 1 ( t ) and u 2 ( t ) for various values of ν and γ with the exact solutions for M = 8, N = 1.…”