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In this paper, we use the method of lines to convert elliptic and hyperbolic partial differential equations (PDEs) into systems of boundary value problems and initial value problems in ordinary differential equations (ODEs) by replacing the appropriate derivatives with central difference methods. The resulting system of ODEs is then solved using an extended block Numerov-type method (EBNUM) via a block unification technique. The accuracy and speed advantages of the EBNUM over the finite difference method (FDM) are established numerically.

The method of lines approach which involves replacing the spatial derivatives with finite difference approximations is commonly used for solving PDEs; whereby, the PDE is transformed into systems of ODEs and solved by reliable ODE solvers (see Lambert [

subject to Dirichlet or Neumann boundary conditions, where

We invoke the method of lines approach in which for real numbers

We then define

The problem (1) then leads to the resulting semi-discrete problem

which can be written in the form

subject to the boundary conditions

where

The paper is organized as follows. In Section 2, we derive a continuous linear multistep method (LMM) which is used to formulate the EBNUM. The computational aspects of the method are given in Section 3. Numerical examples are given in Section 4 to show the accuracy of the method. Finally, the conclusion of the paper is discussed in Section 5.

In this section, we derive a continuous representation of a LMM which is used to generate the EBNUM. On the interval

where

If the function (4) satisfies the scalar form of the differential Equation (3) at the points

Thus, Equations (5) and (6) lead to a system involving the following five equations

which is solved with the aid of Mathematica to obtain

where

The EBNUM is then obtained by evaluating (7) at

Remark 1 We note that the first two members of (9) were given in [

The order of each method in (9) is given by the vector

The method (9) can be expressed in block form as

where the positive integer

given by the coefficients of (9).

Let the local truncation error be defined by

where

Theorem 2 Let

Proof. See Jator [

The linear-stability of (11) is discussed by applying the method to the test equation

where the matrix

Definition 3 Let

Remark 4 We found that

which is twice the stability interval of the standard Numerov method. In this case, the interval of stability is the same as the interval of periodicity.

Recall that the semi-discretization of (1) is initially performed on the partition

Next, we summarize the block unification algorithm. Let

Step 1: Use the block extension of (11) for

Step 2: The blocks are unified to form a system given by

Step 3: The solution of (1) is approximated by the solutions in step 2 as

We emphasize that the block unification technique leads to a single matrix of finite difference equations, which is solved to provide all the solutions of (1) on the entire grid given by the rectangle

In this subsection, the performance of the EBNUM is tested on five problems, which include the Poisson equation, Laplace equation subject to Neumann boundary conditions, Laplace equation subject to Dirichlet boundary conditions, Helmoltz equation, and the two-dimensional convection diffusion equation. In all the figures, the EBNUM is represented by uapprox and the exact solution is represented by uexact.

Example 5 As our first test example, we solve the given Poisson equation (see Burden and Faires [

The exact solution is given by

This example was chosen to demonstrate that the EBNUM can be used to solve the Poisson equation with Dirichlet boundary conditions. The results produced by the EBNUM are accurate as shown by the graphical evidence given in

Example 6 As our second test example, we solve the given Laplace equation subject to Neumann boundary conditions (see Zill and Cullen [

The exact solution is given by

This example was chosen to illustrate that the EBNUM is cable of solving the Laplace equation with Neumann boundary conditions. The results produced by the EBNUM are accurate as shown by the graphical evidence given in

Example 7 As our third test example, we solve the given Laplace equation subject to Dirichlet boundary conditions (see Zill and Cullen [

The exact solution is given by

This example was chosen to demonstrate the performance of the EBNUM on the Laplace equation with Dirichlet boundary conditions. We truncated the exact solution at 50, since the exact solution is an infinite series. The results produced by the EBNUM are accurate as shown by the graphical evidence given in

Example 8 We consider the given two-dimensional Helmoltz equation (see Cheney [

The exact solution is given by

The Dirichlet boundary conditions are chosen accordingly. This example was chosen to demonstrate that the EBNUM can be used to solve the Helmoltz equation. The results produced by the EBNUM are accurate as shown by the graphical evidence given in

Example 9 We consider the given two-dimensional convection diffusion equation (see Sun and Zhang [

The exact solution is given by

The Dirichlet boundary conditions and

Example 10 We consider the following one-dimensional nonlinear undamped Sine-Gordon equation given in Dehghan and Shokri [

The exact solution is given by

C is the velocity of the solitary wave, and the boundary conditions are given according. The problem is solved for

Example 11 We consider the given Telegraph equation (see Ding et al. [

The exact solution is given by

The boundary conditions are chosen accordingly. This example was chosen to demonstrate that the EBNUM can be used to solve the telegraph equation. The results produced by the EBNUM are accurate as shown by the graphical evidence given in

In this subsection, we compare the errors

A block unification method based on the EBNUM is proposed and applied to elliptic and hyperbplic PDEs via the method of lines technique. It is shown that the method is very flexible as it can be applied to solve a variety of elliptic and hyperbolic PDEs with either Dirichlet or Neumann boundary conditions. The method is also shown to have both accuracy and speed advantages when compared with the FDM (see

FDM | EBNUM | ||||
---|---|---|---|---|---|

Example | Error | Time | Error | Time | |

4.1 | 0.41 | 0.38 | |||

4.2 | 0.73 | 0.67 | |||

4.3 | 1.70 | 1.52 | |||

4.4 | 0.44 | 0.34 | |||

4.5 | 0.47 | 0.44 |

search will be to search for higher order LMMs that can solve the general forms of elliptic and hyperbolic PDEs.