Bernoulli Wavelet Method for Numerical Solution of Fokker-Planck-Kolmogorov Time Fractional Differential Equations

The purpose of this paper is to present a wavelet method for numerical solutions Fokker-Planck-Kolmogorov time-fractional differential equations with initial and boundary conditions. The authors was employed the Bernoulli wavelets for the solution of Fokker-Planck-Kolmogorov time-fractional differential equation. We calculated the Bernoulli wavelet fractional integral operation matrix of the fractional order and the upper error boundary for the Riemann‐Levilleville fractional integral operation matrix and the Bernoulli wavelet fractional integral operation matrix. The Fokker-Planck-Kolmogorov time-fractional differential equation is converted to the linear equation using the Bernoulli wavelet operation matrix in this technique. This method has the advantage of being simple to solve. The simulation was carried out using MATLAB software. Finally, the proposed strategy was used to solve certain problems. the Bernoulli wavelet and Bernoulli fraction of the fractional order, the Bernoulli polynomial, and the Bernoulli fractional functions were introduced. Explaining how functions are approximated by fractional-order Bernoulli wavelets as well as fractional-order Bernoulli functions. The Bernoulli wavelet fractional integral operational matrix was used to solve the Fokker-Planck-Kolmogorov fractional differential equations. The results for some numerical examples are documented in table and graph form to elaborate on the efficiency and precision of the suggested method. The results revealed that the suggested numerical method is highly accurate and effective when used to Fokker-Planck-Kolmogorov time fraction differential equations