In this section, we apply HPSTM to nonlinear time-space fractional coupled systems with initial conditions.

Here we apply HPSTM to solve the following nonlinear time-space fractional coupled Burgers system:

Using HPSTM, when [alpha] = [gamma] = [beta] = [delta] = 1, the third approximate solution of (23) is

Under these special conditions, via HPSTM, the first approximate solution of (38) is [mathematical expression not reproducible] (52)

In this paper, we apply the HPSTM to the nonlinear timespace fractional coupled equations.

The numerical results for the time-fractional biological population model (31) obtained by using the HPSTM, SDM, and the exact solution for various values of t, x, and [alpha] with y = 1 and h = 1 are shown by Figures 1(a)-1(d) and those for various values of t at x = 1, y = 1, h = 1, and [alpha] = 1 are depicted in Figure 2 and those for different values of t and [alpha] at x = 1, y = 1, and h = 1 are shown in Figure 3.

which is the same solution as obtained by employing HPSTM and setting [alpha] = 1; it converses to the exact solution U(x, y, t) = [square root of sin x sinh y][e.sup.t].

The numerical results for the time-fractional biological population model (45) obtained with the help of HPSTM, SDM, and the exact solution for various values of t, x, and [alpha] with y = 1 are described through Figures 4(a)-4(d) and those for various values of t at x = 1, y = 1, and [alpha] = 1 are given in Figure 5 and those for different values of t and [alpha] at x = 1 and y = 1 are depicted in Figure 6.

which is the same solution as obtained by the application of HPSTM and setting [alpha] = 1; it converges to the exact solution U(x, y, t) = [e.sup.(1/3)(x+y)-t].

The numerical results for the time-fractional biological population model (55) obtained with the help of HPSTM, SDM, and the exact solution for various values of t, x, and [alpha] with y = 1 are described through Figures 7(a)-7(d) and those for various values of t at x = 1, y = 1, and [alpha] = 1 are depicted in Figure 8 and those for different values of t and [alpha] at x = 1 and y = 1 are presented in Figure 9.