A semi-analytic and numerical approach to the fractional differential equations

A class of linear and nonlinear fractional differential equations (FDEs) in the Caputo sense is considered and studied through two novel techniques called the Homotopy analysis method (HAM). A reliable approach is proposed for solving fractional order nonlinear ordinary differential equations, and the Haar wavelet technique (HWT) is a numerical approach for both integer and noninteger orders. Perturbation techniques are widely applied to gain analytic approximations of nonlinear equations. However, perturbation methods are essentially based on small physical parameters (called perturbation quantity), but unfortunately, many nonlinear problems have no such kind of small physical parameters at all. HAM overcomes this, and HWT does not require any parameters. Due to this, we opt for HAM and HWT to study FDEs. We have drawn a semi-analytical solution in terms of a series of polynomials and numerical solutions for FDEs. First, we solve the models by HAM by choosing the preferred control parameter. Second, HWT is considered. Through this technique, the operational matrix of integration is used to convert the given FDEs into a set of algebraic equation systems. Four problems are discussed using both techniques. Obtained results are expressed in graphs and tables. Results on convergence have been discussed in terms of theorems.