Example 1
Let us consider the following time-fractional initial boundary value problem:
The exact solution, for the special case , is given by
Now we can apply series (4) to construct the solution for , and we have
which is exactly the same solution as in [18].
Example 2
Let us consider the following time-fractional initial boundary value problem:
The exact solution, for the special case , is given by
Now we can apply series (4) to for , and we have
which is exactly the same solution as in [19].
Example 3
Let us consider the nonlinear time-fractional Fisher’s equation
The exact solution, for the special case , is given by
As in the previous examples, we can apply series (4) to obtain the solution for , and we get
which is totaly the same solution as in [20].
Example 4
Let us consider the following time-fractional initial boundary value problem:
(17)
(18)
where the boundary conditions are given in fractional terms.
Boundary value problem (17)-(18), for the special case , becomes as follows:
and its analytic solution is obtained as follows:
Now we can apply series (4) to for , and we have
which is exactly the same solution as in [21].
Example 5
Let us consider the following space-fractional initial boundary value problem:
(19)
(20)
The exact solution, for the special case , is given by
Now we can apply series (9) to for , then we have
Example 6
Let us consider the following space and time-fractional initial boundary value problem:
(21)
(22)
The exact solution, for the special case and , is given by
Now we can apply the series (4)-(9) to for and , then we have
Example 7
Let us consider the following space and time-fractional initial boundary value problem:
(23)
(24)
The exact solution, for the special case and , is given by
Now we can apply the series (4)-(9) to for and , then we have
Figures 7-9 show the evolution results for the approximate solutions of problem (23)-(24) obtained for different values of α and β.