# Numerical Solution of Fokker-Planck-Kolmogorov Time Fractional Differential Equations Using Legendre Wavelet Method Along with convergence and error analysis

نوع مقاله : مقاله پژوهشی

نویسندگان

دانشکده علوم ریاضی، دانشگاه صنعتی شاهرود، سمنان، ایران

چکیده

The purpose of this paper is to present an efficient numerical method for finding numerical solutions Fokker-Planck-Kolmogorov time-fractional differential equations.The Legendre wavelet approach was employed for this objective. The Legendre wave was the first to be introduced. The Fokker-Planck-Kolmogorov time-fractional differential equation is converted to the linear equation using the Legendre 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 absolute value of the error between the precise and approximate answers provided by the numerical technique was then introduced, and the numerical method's error was analyzed.The results revealed that the suggested numerical method is highly accurate and effective when used to Fokker-Planck-Kolmogorov time fraction 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. Moreover, for the convergence of the proposed technique, inequality is derived in the context of error analysis. In this paper and for the first time: the authors presented results on the numerical simulation for classes of time-fractional differential equations. The authors applied the reproducing Legendre wavelet method for the numerical solutions of nonlinear Fokker-Planck-Kolmogorov time-fractional differential equation.The method presented in the present study can be used by programmers, engineers and other researchers in this field.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

### Numerical Solution of Fokker-Planck-Kolmogorov Time Fractional Differential Equations Using Legendre Wavelet Method Along with convergence and error analysis

نویسندگان [English]

• S. Reza Hejazi
Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Semnan, Iran.
چکیده [English]

The aim of this paper is to numerically solve the Fokker-Planck-Kolmogorov fractional-time differential equations using the Legendre wavelet. Also, we analyzed the convergence of function approximation using Legendre wavelets. Introduced the absolute value between the exact answer and the approximate answer obtained by the given numerical methods, and analyzed the error of the numerical method. This method has the advantage of being simple to solve. The results revealed that the suggested numerical method is highly accurate and effective. 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 simulation was carried out using MATLAB software. In this paper and for the first time, the authors presented results on the numerical simulation for classes of time-fractional differential equations. The authors applied the reproducing Legendre wavelet method for the numerical solutions of nonlinear Fokker-Planck-Kolmogorov time-fractional differential equation.The method presented in the present study can be used by programmers, engineers and other researchers in this field.

کلیدواژه‌ها [English]

• Fokker-Planck-Kolmogorov
• differential equations
• Legendre wavelet
• fractional integration

#### مراجع

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### سابقه مقاله

• تاریخ دریافت: 15 اسفند 1400
• تاریخ بازنگری: 31 اردیبهشت 1401
• تاریخ پذیرش: 27 آذر 1402