Application 1

Consider the following linear fractional-order 2-by-2 stiff system:

with the initial conditions

where and are constants. To obtain the solution of (4.1) by NHPM, we construct the following homotopy:

Applying the inverse operator, of both sides of the above equation, we obtain

The solution of (4.1) to has the following form:

Substituting (4.5) in (4.4) and equating the coefficients of like powers of , we get the following set of equations:

Assuming , , , , and and solving the above equation for and lead to the result

Vanishing and lets the coefficients by taking the following values:

Therefore, we obtain the solutions of (4.1) as

Our aim is to study the mathematical behavior of the solution and for different values of . This goal can be achieved by forming Pade' approximants, which have the advantage of manipulating the polynomial approximation into a rational function to gain more information about and . It is well known that Pade' approximants will converge on the entire real axis, if and are free of singularities on the real axis. It is of interest to note that Pade' approximants give results with no greater error bounds than approximation by polynomials. To consider the behavior of solution for different values of , we will take advantage of the explicit formula (4.9) available for and consider the following two special cases.

Case 1.

Setting in (4.9), we obtain the approximate solution in a series form as

Case 2.

In this case, we will examine the linear fractional stiff equation (4.1). Setting in (4.9) gives

For simplicity, let , then

Calculating the [10/11] Pade' approximants and recalling that , we get

Application 2

Consider the following nonlinear fractional-order 2-by-2 stiff system:

with the initial conditions

To obtain the solution of (4.14) by NHPM, we construct the following homotopy:

Applying the inverse operator, of both sides of the above equation, we obtain

The solution of (4.14) to have the following form:

Substituting (4.18) in (4.17) and equating the coefficients of like powers of , we get the following set of equations:

Assuming , , , , and and solving the above equation for and lead to the result

Vanishing and lets the coefficients to take the following values:

Therefore, we obtain the solution of (4.14) as

The exact solution of (4.14) for is .

Application 3

Consider the following nonlinear Genesio system with fractional derivative:

with the initial conditions

where , , and are constants. To obtain the solution of (4.23) by NHPM, we construct the following homotopy:

Applying the inverse operator, of both sides of the above equation, we obtain

The solution of (4.23) to have the following form:

Substituting (4.27) in (4.26) and equating the coefficients of like powers of , we get the following set of equations:

Assuming , , , , , , , , and , and solving the above equation for and lead to the result

Vanishing , and lets the coefficients to take the following values:

Therefore, we obtain the solutions of (4.23) as

Application 4

Finally, we consider the following nonlinear matrix Riccati differential equation with fractional derivative:

where . To find the solution of this equation by NHPM, we will treat the matrix equation as a system of fractional-order differential equations

with the initial conditions

Therefore, we obtain the solution of (4.33) as