Abstract
This research focuses on the approximate solutions of second-order fuzzy differential equations with fuzzy initial condition with two different methods depending on the properties of the fuzzy set theory. The methods in this research based on the Optimum homotopy asymptotic method (OHAM) and homotopy analysis method (HAM) are used implemented and analyzed to obtain the approximate solution of second-order nonlinear fuzzy differential equation. The concept of topology homotopy is used in both methods to produce a convergent series solution for the propped problem. Nevertheless, in contrast to other destructive approaches, these methods do not rely upon tiny or large parameters. This way we can easily monitor the convergence of approximation series. Furthermore, these techniques do not require any discretization and linearization relative with numerical methods and thus decrease calculations more that can solve high order problems without reducing it into a first-order system of equations. The obtained results of the proposed problem are presented, followed by a comparative study of the two implemented methods. The use of the methods investigated and the validity and applicability of the methods in the fuzzy domain are illustrated by a numerical example. Finally, the convergence and accuracy of the proposed methods of the provided example are presented through the error estimates between the exact solutions displayed in the form of tables and figures.
Highlights
As a mathematical model, a large number of dynamic real-life problems can be formulated in mathematical equations
Fuzzy differential equations with fuzzy initial conditions appear when the modeling of these problems was imperfect and its nature is under uncertainty that involves fuzzy parameters that cannot be detected through ordinary measurement [1]
Some approximate methods are involved with an approximate solution of various types and order of fuzzy differential equations by Nedal et al [7] and Jameel et al [8]. Both homotopy analysis method (HAM) and Optimum homotopy asymptotic method (OHAM) are classified as approximate methods that have been used to solve differential equations approximately in various applications [9,10,11,12,13,14] that have many advantages such as solving the difficult nonlinear
Summary
A large number of dynamic real-life problems can be formulated in mathematical equations. Some approximate methods are involved with an approximate solution of various types and order of fuzzy differential equations by Nedal et al [7] and Jameel et al [8] Both HAM and OHAM are classified as approximate methods that have been used to solve differential equations approximately in various applications [9,10,11,12,13,14] that have many advantages such as solving the difficult nonlinear. Approximate methods like HAM and OHAM provide a simple way to ensure the convergence of a series solution that comes from the great freedom to choose proper base function approximating a nonlinear problem [15]. Our main motivation is to present a complete fuzzy analysis of the problem of second-order fuzzy initials, followed by a fuzzy analysis of OHAM and HAM, to obtain an approximate solution to the proposed problem and to present a comparative study of these methods in detail. There is a summary that contains the conclusions of this paper
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