Semi-analytic solutions of nonlinear multidimensional fractional differential equations

Area of Science:

• Mathematics
• Applied Mathematics
• Numerical Analysis

Background:

• Fractional differential equations (FDEs) are crucial in modeling complex phenomena.
• The Caputo-Fabrizio fractional derivative offers advantages with its non-singular kernel.
• Solving nonlinear multidimensional FDEs presents significant analytical and numerical challenges.

Purpose of the Study:

• To apply and compare the Adomian decomposition method (ADM) and Picard technique for solving nonlinear multidimensional FDEs with the Caputo-Fabrizio fractional derivative.
• To establish conditions for the existence and uniqueness of solutions.
• To analyze the convergence and estimate the maximum absolute error of the obtained series solutions.

Main Methods:

• The Adomian decomposition method (ADM) is employed to derive series solutions.
• The Picard technique is utilized for iterative approximation of solutions.
• Both methods are applied to nonlinear multidimensional FDEs featuring the Caputo-Fabrizio fractional derivative.

Main Results:

• A sufficient condition for the existence and uniqueness of a solution is derived.
• The convergence of the series solutions obtained by both methods is discussed.
• Numerical examples, including those without exact solutions, are solved and compared.
• The Adomian decomposition method demonstrates significantly lower computational time compared to the Picard technique.

Conclusions:

• Both ADM and Picard technique are effective for solving the studied class of FDEs.
• The Caputo-Fabrizio fractional derivative provides a robust framework for modeling.
• ADM offers a computationally efficient alternative to the Picard technique for these problems.