Optimization of the solution of the Fisher stochastic-financial equation by implementing the colocal numerical expansion method with orthogonal bases

Subject Areas : Financial Economics

شادان صدیق بهزادی ^{1 *} , محمد علوی ششتمد ²

1 - هیات علمی گروه ریاضی و آمار دانشگاه آزاد اسلامی واحد قزوین
2 - هیات علمی گروه مدیریت بازرگانی دانشگاه آزاد اسلامی واحد تهران مرکزی

Received: 2022-05-21 Published : 2025-09-06

Keywords: JEL : C02 , C22,

Abstract :

Abstract

In this paper, we solve the Fisher equation by the numerical collocation method with the orthogonal Jacobi and airfoil bases. This partial differential equation is one of the most important and widely used equations in financial mathematics and economics. The Fisher equation shows how inflation expectations affect interest rates and purchasing power. This equation offers a solution for quadratic partial differential equations and stochastic differential equations. Applications of this equation in the field of random control, mathematical financing, risk analysis and related fields can be mentioned. In this paper, by implementing numerical methods on the desired equation, nonlinear devices are obtained that can they can be solved by methods for solving nonlinear devices, such as Newton's iterative method. The existence, uniqueness of the solution and convergence of the methods are examined and in an example we will show that with a low number of repetitions and errors and high convergence speed we converge to the approximate solution of the equation and this shows the optimality of this numerical method compared to analytical methods.

Keywords: Collocation method,Jacobi polynomial, Airfoil polynomial, Fisher equation.

References:

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