Determining the optimal response of Feynman-Kak stochastic-financial equation based on Jacobi and Irfoil expansion

In this paper, we solve the Feynman-Kac equation using the collocation method with Jacobi and Airfoil bases. This partial differential equation is one of the most important and widely used random equations in financial mathematics. Due to the increasing demand in applied sciences such as financial mathematics, economics and complexity in modeling, data analysis and calculation, significant efforts have been made in search of better mathematical models to obtain approximate solutions to the modeled equations in recent years. It is well established that many of the systems encountered in the new era cannot be represented by ordinary differential equations in the traditional way or by the model of random differential equations. This equation offers a solution for quadratic partial differential equations and stochastic differential equations. Applications of this formula in the field of random control, financial mathematics, risk analysis and related fields can be mentioned. In this paper, by applying numerical methods to the Feynman-Kac equation, nonlinear devices are obtained that can be solved by numerical methods for solving nonlinear devices, such as Newton's iterative method.