Topological Degree Theory to Investigate Fractional Order Korteweg-de Vries Equation of Porous Media

The Korteweg-de Vries (KdV) equation is considered in this manuscript 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 regards, we considered a general form of 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 surfaces plots have presented to investigate the physical impact of fractional order, time and space variable on the propagation of wave solution.