^{1}

^{2,3,4}

^{1}

^{2}

^{3}

^{4}

We solve the system of nonlinear fractional Jaulent-Miodek and Whitham-Broer-Kaup equations via the Sumudu transform homotopy method (STHPM). The method is easy to apply, accurate, and reliable.

Nonlinear partial differential equations arise in various areas of physics, mathematics, and engineering [

The nonlinear FWBK equation which will be considered in this paper has the following form:

The paper is organized as follows. In Section

A real function

The Riemann-Liouville fractional integral operator of order

For

The Caputo fractional order derivative is given as follows [

The Riemann-Liouville fractional order derivative is given as follows [

The Jumarie Fractional order derivative is given as follows [

If

Assume now that

The Sumudu transform of a function

Let

there exist a circular region

For the proof see [

We illustrate the basic idea of this method [

Applying the Sumudu transform on both sides of (

Now we apply the following HPM:

In this section, we apply this method for solving the system of the fractional differential equation. We will start with (

Following carefully the steps involved in the STHPM, after comparing the terms of the same power of

Approximate solution for FWBK equation.

Approximate solution of FWBK equation.

Approximate solution of FWBK equation.

Approximate solution of FWBK equation.

For (

Approximate solution of FJM equation.

Approximate solution of FJM equation.

We derived approximated solutions of nonlinear fractional Jaulent-Miodek and Whitham-Broer-Kaup equations using the relatively new analytical technique the STHPM. We presented the brief history and some properties of fractional derivative concept. It is demonstrated that STHPM is a powerful and efficient tool for the system of FPDEs. In addition, the calculations involved in STHPM are very simple and straightforward.

The STHPM is chosen to solve this nonlinear problem because of the following advantages that the method has over the existing methods. This method does not require the linearization or assumptions of weak nonlinearity. The solutions are not generated in the form of general solution as in the Adomian decomposition method (ADM) [

^{+}cells and attractor one-dimensional Keller-Segel equation