^{1}

^{2,3}

^{4}

^{1}

^{2}

^{3}

^{4}

A Jacobi dual-Petrov-Galerkin (JDPG) method is introduced and used for solving fully integrated reformulations of third- and fifth-order ordinary differential equations (ODEs) with constant coefficients. The reformulated equation for the

A well-known advantage of spectral methods is high accuracy with relatively fewer unknowns when compared with low-order finite-difference methods [

Third-order differential equations have applications in many engineering models, see for instance [

In this paper, the proposed differential equations are integrated

The main aim of this paper is to propose a suitable way to approximate some integrated forms of third- and fifth-order ODEs with constant coefficients using a spectral method, based on the Jacobi polynomials such that it can be implemented efficiently and at the same time has a good convergence property. It is worthy to note here that odd-order problems lack the symmetry of even-order ones, so we propose a Jacobi dual-Petrov-Galerkin (JDPG) method. The method leads to systems with specially structured matrices that can be efficiently inverted. We apply the method for solving the integrated forms of third- and fifth-order ODEs by using compact combinations of the Jacobi polynomials, which satisfy essentially all the underlying homogeneous boundary conditions. To be more precise, for the JDPG we choose the trial functions to satisfy the underlying boundary conditions of the differential equations, and we choose the test functions to satisfy the dual boundary conditions. Extension of the JDPG for polynomial coefficient ODEs is obtained by approximating the weighted inner products in the JDPG by using the Jacobi-Gauss-Lobatto quadrature. Finally, examples are given to illustrate the efficiency and implementation of the method. Comparisons are made to confirm the reliability of the method.

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

Let

Let

For any

For any real numbers

The following special values will be of fundamental importance in what follows (see, [

If we define the

One has

If one writes

It is immediately obtained from relation (

We are interested in using the JDPG method to solve the third-order differential equation

We choose compact combinations of the Jacobi polynomials as basis functions to minimize the bandwidth hoping to improve the condition number of the coefficient matrix corresponding to (

Now it is clear that (

Hence by setting

In the case of

It is worthy to note that, for

The JDPG can be extended for ODEs with polynomial coefficients because of analytical form of a product of an algebraic polynomial, and the Jacobi polynomials are known.

Now the formula of the Jacobi coefficients of the moments of one single Jacobi polynomial of any degree (see, Doha [

Let us consider the following integrated form of the third-order differential equation:

Let us consider

In this section, we consider the fifth-order differential equation of the form

Here, we apply the dual-Petrov-Galerkin approximation to (

We set

We choose the coefficients

In this subsection, we consider the fifth-order differential equation (

Equation (

In this section some examples are considered aiming to illustrate how one can apply the proposed algorithms presented in the previous sections. Comparisons between JDPG method and other methods proposed in [

Consider the one-dimensional third-order equation

Table

Maximum pointwise error using JDPG method for

JDPG | JDPG | |||||
---|---|---|---|---|---|---|

8 | ||||||

16 | ||||||

24 | ||||||

8 | ||||||

16 | ||||||

24 |

Consider the one-dimensional fifth-order differential problem

Table

Maximum pointwise error using JDPG method for

JDPG | JDPG | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

12 | ||||||||||

24 | 1 | 1 | 1 | |||||||

36 | ||||||||||

12 | ||||||||||

24 | 0 | 0 | 1 | 1 | 1 | |||||

36 | ||||||||||

12 | ||||||||||

24 | 1 | 1 | 1 | |||||||

36 |

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

Maximum pointwise error of

Our algorithm | In [ | In [ | In [ | In [ | |||
---|---|---|---|---|---|---|---|

10 | 0 | 0 | 0.1570 | ||||

13 | 0 | 0 | |||||

20 | 0 | 0 | 0.0747 | ||||

26 | 0 | 0 | |||||

Consider the following third-order ODE with polynomial coefficients:

Equation (

Using the quadrature dual-Petrov-Galerkin method described in Section

Maximum pointwise error using quadrature JDPG method for

Quadrature JDPG | Quadrature JDPG | |||||
---|---|---|---|---|---|---|

8 | 0 | 0 | ||||

12 | ||||||

16 | ||||||

20 |

In this paper, we described a JDPG method for fully integrated forms of third- and fifth-order ODEs with constant coefficients. Because of the constant coefficients, the matrix elements of the discrete operators are provided explicitly, and this in turn greatly simplifies the steps and the computational effort for obtaining solutions. However, the integrated form of the source function (involving severalfold indefinite integrals) should be known analytically, and the right hand side vector require quadrature approximations. This approach is also considered for ODEs with polynomial coefficients. Numerical results exhibit the high accuracy of the proposed numerical methods of solutions.

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