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We have presented an efficient spectral algorithm based on shifted Jacobi tau method of linear fifth-order two-point boundary value problems (BVPs). An approach that is implementing the shifted Jacobi tau method in combination with the shifted Jacobi collocation technique is introduced for the numerical solution of fifth-order differential equations with variable coefficients. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplify the problem. Shifted Jacobi collocation method is developed for solving nonlinear fifth-order BVPs. Numerical examples are performed to show the validity and applicability of the techniques. A comparison has been made with the existing results. The method is easy to implement and gives very accurate results.

The solutions of fifth-order BVPs have been the subject of active research. These problems generally arise in mathematical modeling of viscoelastic flows, physics, engineering, and other disciplines, (see, e.g., [

Recently, various powerful mathematical methods such as the sixth-degree B-spline [

Spectral methods (see, e.g., [

The use of general Jacobi polynomials has the advantage of obtaining the solutions of differential equations in terms of the Jacobi parameters

In the tau method (see, e.g., [

Since 1960 the nonlinear problems have attracted much attention. In fact, there are many analytic asymptotic methods that have been proposed for addressing the nonlinear problems. Recently, the collocation method [

The fundamental goal of this paper is to develop a direct solution algorithm for approximating the linear two-point fifth-order differential equations by shifted Jacobi tau (SJT) method that can be implemented efficiently. Moreover, we introduce the pseudospectral shifted Jacobi tau (P-SJT) method in order to deal with fifth-order BVPs of variable coefficients. This method is basically formulated in the shifted Jacobi tau spectral form with general indexes

For nonlinear fifth-order problems on the interval

The remainder of this paper is organized as follows. In Section

Let

It is well known that

Now, let

We denote by

Let

The

In this section, we are intending to use the SJT method to solve the fifth-order boundary value problems

Let us first introduce some basic notation that will be used in this section. We set

If we assume that

Then, recalling (

In this section, we use the pseudospectral shifted Jacobi tau (P-SJT) method to numerically solve the following fifth-order boundary value problem with variable coefficients

We define the discrete inner product and norm as follows:

Obviously,

The pseudospectral tau method for (

Hence, by setting

In this section, we are interested in solving numerically the nonlinear fifth-order boundary value problem

In this section, we apply shifted Jacobi tau (SJT) method for solving the fifth-order boundary value problems. Numerical results are very encouraging. For the purpose of comparison, we took the same examples as used in [

Consider the following linear boundary value problem of fifth-order (see [

Table

Absolute errors using SJT method for

SJT method | VIMHP | B-spline | VIM | ADM | ITM | HPM | |||
---|---|---|---|---|---|---|---|---|---|

0.0 | 0 | 0 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |

0.2 | 0 | 0 | |||||||

0.4 | 0 | 0 | |||||||

0.6 | 0 | 0 | |||||||

0.8 | 0 | 0 | |||||||

1.0 | 0 | 0 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |

Consider the following nonlinear boundary value problem of fifth-order (see [

In Table

Absolute errors using SJC method for

SJC method | VIMHP | B-spline | VIM | ADM | ITM | HPM | |||
---|---|---|---|---|---|---|---|---|---|

0.0 | 0 | 0 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |

0.2 | 0 | 0 | |||||||

0 | |||||||||

0.4 | 0 | 0 | |||||||

0.6 | 0 | 0 | |||||||

0.8 | 0 | 0 | |||||||

1.0 | 0 | 0 | 0 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

0 |

In this paper, we have presented some efficient direct solvers for the general fifth-order BVPs by using Jacobi-tau approximation. Moreover, we developed a new approach implementing shifted Jacobi tau method in combination with the shifted Jacobi collocation technique for the numerical solution of fifth-order BVPs with variable coefficients. Furthermore, we proposed a numerical algorithm to solve the general nonlinear fifth-order differential equations by using Gauss-collocation points and approximating directly the solution using the shifted Jacobi polynomials. The numerical results in this paper demonstrate the high accuracy of these algorithms.

The authors are very grateful to the referees for carefully reading the paper and for their comments and suggestions which have improved the paper.