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Solving two-dimensional nonlinear fuzzy Volterra integral equations by homotopy analysis method

Georgieva Atanaska

Demonstratio Mathematica

2021

Vol. 54, nr 1

11--24

The purpose of the paper is to find an approximate solution of the two-dimensional nonlinear fuzzy Volterra integral equation, as homotopy analysis method (HAM) is applied. Studied equation is converted to a nonlinear system of Volterra integral equations in a crisp case. Using HAM we find approximate solution of this system and hence obtain an approximation for the fuzzy solution of the nonlinear fuzzy Volterra integral equation. The convergence of the proposed method is proved. An error estimate between the exact and the approximate solution is found. The validity and applicability of the HAM are illustrated by a numerical example.

Two-step collocation methods for two-dimensional Volterra integral equations of the second kind

Torabi Seyed Mousa, Tari Abolfazl, Shahmorad Sedaghat

Journal of Applied Analysis

2019

Vol. 25, nr 1

1--11

In this paper, we develop two-step collocation (2-SC) methods to solve two-dimensional nonlinear Volterra integral equations (2D-NVIEs) of the second kind. Here we convert a 2D-NVIE of the second kind to a one-dimensional case, and then we solve the resulting equation numerically by two-step collocation methods. We also study the convergence and stability analysis of the method. At the end, the accuracy and efficiency of the method is verified by solving two test equations which are stiff. In examples, we use the well-known differential transform method to obtain starting values.