TOPOLOGICAL DEGREE THEORY TO INVESTIGATE FRACTIONAL ORDER KORTEWEG–DE VRIES EQUATION OF POROUS MEDIA

Abstract

The Korteweg–de Vries (KdV) equation is considered in this paper with fractional order derivative. We have considered the aforesaid equation under the non-singular derivative of fractional order introduced by Atangana–Baleanu–Caputo (ABC). We have implemented topological degree theory to deduce sufficient results for the existence and uniqueness of solution to the concerned problem. The concerned degree theory has been used to study a class of nonlinear integral equation. In this regard, we considered a general form for our proposed problem and then established appropriate conditions required for the existence and uniqueness of solution. Additionally, stability is an important requirement in investigating analytical or numerical solutions, therefore, we have attempted to derive some adequate results for the Hyers–Ulam-type stability. Finally, for computing the approximate solution, we have used Laplace transform coupled with Adomian decomposition (LADM) to study the mentioned problem. Two concerted examples have been testified by the mentioned technique. Different surface plots have been presented to investigate the physical impact of fractional order, time and space variable on the propagation of wave solution.