Boundary conditions; Boundary value problems; Fuzzy systems; Inverse problems; Linear equations; Numerical methods; Partial differential equations, Black Scholes equations; Crank-Nicolson scheme; Dirichlet boundary condition; Inverse F-transforms; Numerical solution of partial differential equations; Partial Differential Equations (PDEs); System of linear equations; Two-dimensional boundary value problems, Mathematical transformations

In this paper, we propose a numerical method based on the F-transform for solving a certain type of partial differential equations (PDEs) with Dirichlet boundary conditions and initial conditions. We show how the PDEs after the application of the Crank-Nicolson scheme for time discretization can be approximated by a system of linear equations with the direct F-transform components as their variables. The numerical solution of PDEs is obtained by solving the system of linear equations and the application of the inverse F-transform. The proposed method is then adjusted to the two-dimensional boundary value problem illustrated on two examples including the Black-Scholes equation well-known in financial modeling.

In this paper, we propose a numerical method based on the F-transform for solving a certain type of partial differential equations (PDEs) with Dirichlet boundary conditions and initial conditions. We show how the PDEs after the application of the Crank-Nicolson scheme for time discretization can be approximated by a system of linear equations with the direct F-transform components as their variables. The numerical solution of PDEs is obtained by solving the system of linear equations and the application of the inverse F-transform. The proposed method is then adjusted to the two-dimensional boundary value problem illustrated on two examples including the Black-Scholes equation well-known in financial modeling.