A Novel Method for Solving Riemann–Liouville Fractional Differential Equations

Riemann–Liouville initial value problems are well‐posed for wide class of functions including those which may not be differentiable at the initial point. In the present paper, an efficient numerical tool for solving fractional differential equations containing Riemann–Liouville derivative is developed. Newly proposed method is termed as “Mahatekar–Deshpande predictor–corrector method,” in short “MDPCM.” Fundamental concepts of fractional calculus and numerical analysis are used to derive the convergence analysis and stability analysis of the newly proposed method. New method has been formed using a combination of a very famous decomposition tool, namely, “new iterative method” and Crank–Nikolson method. Firstly, formation of the algorithm of the newly proposed method is done, and then, to check the accuracy and efficiency of the new method, numerous examples are solved. Solutions obtained by the new method are compared with exact solutions, and it has been found that the new algorithm works accurately for solving fractional differential equations with Riemann–Liouville derivative.