In this section, we derive the SJ-GL-C method to solve numerically the following model problem:
subject to the three-point boundary conditions
where denotes the Caputo fractional derivative of order ν for λ is a real number, are given constants and is a given nonlinear source function. For the existence and uniqueness of solution of (11)-(12), see .
The choice of collocation points is important for the convergence and efficiency of the collocation method. For boundary value problems, the Gauss-Lobatto points are commonly used. It should be noted that for a differential equation with the singularity at in the interval one is unable to apply the collocation method with Jacobi-Gauss-Lobatto points because the two assigned abscissas 0 and L are necessary to use as a two points from the collocation nodes. Also, a Jacobi-Gauss-Radau nodes with the fixed node cannot be used in this case. In fact, we use the collocation method with Jacobi-Gauss-Lobatto nodes to treat the nonlinear Langevin differential equation; i.e., we collocate this equation only at the Jacobi-Gauss-Lobatto points . These equations together with three-point boundary conditions generate nonlinear algebraic equations which can be solved.
Let us first introduce some basic notation that will be used in the sequel. We set
We next recall the Jacobi-Gauss-Lobatto interpolation. For any positive integer N
stands for the set of all algebraic polynomials of degree at most N
. If we denote by
), to the nodes and Christoffel numbers of the standard (shifted) Jacobi-Gauss-Lobatto quadratures on the intervals
respectively. Then one can easily show that
We introduce the following discrete inner product and norm:
where and are the nodes and the corresponding weights of the shifted Jacobi-Gauss-quadrature formula on the interval respectively.
Due to (14), we have
Thus, for any , the norms and coincide.
Associating with this quadrature rule, we denote by
the shifted Jacobi-Gauss interpolation,
The shifted Jacobi-Gauss collocation method for solving (11)-(12) is to seek
, such that
We now derive an efficient algorithm for solving (17)-(18). Let
We first approximate
, as Eq. (19
). By substituting these approximations in Eq. (11
), we get
Here, the fractional derivative of order μ in the Caputo sense for the shifted Jacobi polynomials expanded in terms of shifted Jacobi polynomials themselves can be represented formally in the following theorem.
Theorem 3.1 Letbe a shifted Jacobi polynomial of degree j, then the fractional derivative of order ν in the Caputo sense foris given by
Proof This theorem can be easily proved (see Doha et al. ).
In practice, only the first
-terms shifted Jacobi polynomials are considered, with the aid of Theorem 3.1 (Eq. (21
)), we obtain from (20) that
Also, by substituting Eq. (19
) in Eq. (12
) we obtain
To find the solution
, we first collocate Eq. (22
) at the
shifted Jacobi-Gauss-Lobatto notes, yields
Next, Eq. (23
), after using (9) and (6), can be written as
The scheme (24)-(25) can be rewritten as a compact matrix form. To do this, we introduce the
with the entries
Also, we define the
with the entries:
with the entries:
is the k
th component of C a
. Then we obtain from (24)-(25) that
Finally, from (26), we obtain nonlinear algebraic equations which can be solved for the unknown coefficients by using any standard iteration technique, like Newton’s iteration method. Consequently, given in Eq. (19) can be evaluated. □
Remark 3.2 In actual computation for fixed μ, ν and λ, it is required to compute only once. This allows us to save a significant amount of computational time.