The new iterative method with a powerful algorithm is developed for the solution of linear and nonlinear ordinary and partial differential equations of fractional order as well. The analysis is accompanied by numerical examples where this method, in solving them, is used without linearization or small perturbation which con

Considerable attention has been devoted to the study of the fractional calculus during the past three decades and its numerous applications in the area of physics and engineering. The applications of fractional calculus used in many fields such as electrical networks, control theory of dynamical systems, probability and statistics, electrochemistry, chemical physics, optics, and signal processing can be successfully modelled by linear or nonlinear fractional differential equations.

So far there have been several fundamental works on the fractional derivative and fractional differential equations [

Finding approximate or exact solutions of fractional differential equations is an important task. Except for a limited number of these equations, we have difficulty in finding their analytical solutions. Therefore, there have been attempts to develop new methods for obtaining analytical solutions which reasonably approximate the exact solutions. Several such techniques have drawn special attention, such as Adomain’s decomposition method [

The motivation of this paper is to extend the application of the new iterative method proposed by Daftardar-Gejji and Jafari [

There are several definitions of a fractional derivative of order

Caputo fractional derivative first computes an ordinary derivative followed by a fractional integral to achieve the desired order of fractional derivative. Riemann-Liouville fractional derivative is computed in the reverse order. Therefore, Caputo fractional derivative allows traditional initial and boundary conditions to be included in the formulation of the problem.

From properties of

For the basic idea of the new iterative method, we consider the following general functional equation [

Now we analyze the convergence of the new iterative method for solving any general functional equation (

In this section, we introduce a suitable algorithm for solving nonlinear partial differential equations using the new iterative method. Consider the following nonlinear partial differential equation of arbitrary order:

When the general functional equation (

From the properties of integration and by using (

To illustrate the effectiveness of the proposed method, several test examples are carried out in this section.

In this example, we consider the following initial value problem in the case of the inhomogeneous Bagely-Torvik equation [

By applying the technique described in Sections

Let

In Example

In Figure

Plots of the approximate solution and the exact solution for (

Comparing these obtained results with those obtained by new Jacobi operational matrix in [

Consider the following fractional Riccati equation [

By applying the technique described in Sections

Let

In Figure

Plots of the approximate solution for different values of

Comparing the obtained results with those obtained by homotopy analysis method, in case

Consider the following initial value problem with fractional order [

The exact solution for this problem is

As in Example

Let

Plots of the approximate solution and the exact solution for (

Comparing these obtained results with those obtained by new Jacobi operational matrix in [

Consider the following fractional order wave equation in

The exact solution for this problem when

The initial value problem (

Let

Consider the following fractional order heat equation in

The exact solution for this problem when

The initial value problem (

Let

The obtained results in this example are the same as these obtained in [

In this last example, we consider the following fractional order nonlinear wave equation [

The exact solution for this problem when

The initial value problem (

Let

(a) Plots of the approximate solution for different values of

Comparing these results with those obtained by the modification homotopy perturbation method in [

In this paper, the new iterative method with suitable algorithm is successfully used to solve linear and nonlinear ordinary and partial differential equations with fractional order. It is clear that the computations are easy and the solutions agree well with the corresponding exact solutions and more accurate than the solutions obtained by other methods. Moreover, the accuracy is high with little computed terms of the solution which confirm that this method with the given algorithm is a powerful method for handling fractional differential equations.