_{1}

^{*}

We present the numerical method for solution of some linear and non-linear parabolic equation. Using idea [1], we will present the explicit unconditional stable scheme which has no restriction on the step size ratio
*k/h*
^{2} where
*k* and h are step sizes for space and time respectively. We will also present numerical results to justify the present scheme.

A number of difference schemes for solving partial difference equations have been proposed by using the idea of

methods of lines [

where k and h are step sizes for space and time respectively. We [

with the initial Dirichlet boundary condition

where we set

We propose the difference approximation to (1.1) where the step size ratio is defined by

(c is any positive constant) (1.4)

The outline of this paper is as follows. In §2, by using idea of methods of lines, we present the explicit difference approximation to (1.1). In §3 we study the truncation errors of our scheme. In §4 we study the convergence of the scheme with the condition (1.4) and we will show that our scheme converges to the true solution of (1.1). In §5 we study stability of the scheme, and we will show that our scheme is stable for any step size k and h with the condition (1.4). In §6 we show some numerical examples to justify our methods.

In the same way as in [

where

We set

We define the difference approximation to (1.1) by the following scheme.

If

Then we set

If

Then we set

where

The step size

If we set

Then, from (2.3), (2.4), we have

We define the truncation error

where, from the definition of (2.7), we have

By Taylor series expansions of the solution

From (3.3), we have

If we set

and

Then, from (3.4), we have the following result.

Theorem 1. The truncation error of the difference approximation (2.8) to (1.1) is given by

where

where

In this section, we study the convergence of the scheme (2.8). We set the approximation error by

We use the abbreviation's

From (2.8), (3.7), (4.1), we have

with

From (2.5), we have

We set the initial conditions of (4.2)

Form (4.2), (4.4), (4.5), (4.6), we have

From (4.7), we have

with

We study the coefficients of (4.8) to

Firstly we consider the case

We set

Then from (4.3), (4.12), we have

We have the equation

From (4.13), (4.14), (4.15), (4.16), we have

If we assume

Then we have

From (3.7), we have

with

From (4.20), we have

where

We have from the condition (1.1)

From (4.17), (4.20), (4.23), we have

where

In the same way to (4.16), from (4.10), we have

From (3.8), we have

After some complicate computation, we have

with

From (4.27), we have

with

From (4.26), we have

with

From (4.30)

with

From (4.26), (4.28), (4.31), we have

From (4.25), we have

From (4.25), (4.33), (4.34), we have

where

From (4.20), we have

where

From (4.8), (4.20) (4.24), (4.35), (4.36), we have

where

We set the maximum norm of the function

Then, from (4.37), we have

From (4.39), we have

Finally we assume

Then, from (4.3), we have

From (4.9), (4.42), we have

In the same way to (4.14), we have

From (3.8), we have after some computation,

with

From (4.8), (4.20), (4.43), (4.44), (4.45), we have

where

Then, in the same way to (4.40), from (4.47), we have

We study l = 0. In the almost same way to (4.47), we have

where C_{1} and C_{4} are defined by (4.21) and (4.46) with l = 0 respectively.

From (4.49), we have

From (4.40), (4.48), (4.50), we have

Theorem 2. Suppose that for

If the solution

In this section, we study the stability of the numerical process (2.8) and define as follows.

Definition: The numerical processes

where

We prove that the scheme (2.8) are stable in mean of the von Neumann.

We set

Then, from (4.7), we have

From (5.1), we have

where

If we assume (4.18) on the solution

for some constant

From (5.2), (5.3), we have the following result.

Lemma 1. If we assume the solution

with

where

We set the maximum norm of the function

We have the inequality

From (1.1), we have

From (5.8), we have

From (2.8), we have

and

From (5.10), (5.11), we have

Firstly we consider

Then from (5.9) and (5.12), we have

with

where K,

From (5.14), we have

Lastly, we consider

From (5.12), we have

Firstly, we consider the case

Then from (5.16), we have

We have

From (5.10). (5.17), (5.18), we have

with

where K and

If

From (5.21), we have

If

From (5.23), we have

From (5.22), (5.24), we set

where

From (5.6), (5.19) and (5.25), we have

and we have the following result

From (5.26), we have

where

Secondly, in the case

From (5.28), we have

with

where K and

In the same way to (5.26), we have

where

From (5.15), (5.27), (5.31), we have

Theorem 3.

If the solution

Lastly, we study the numerical test in the following non-linear Equation .

and the initial and boundary problem given by,

Applying the difference Equation (2.8) to (6.1) with (6.2), we have the the numerical results in

As we see in

MasaharuNakashima, (2015) Unconditionally Explicit Stable Difference Schemes for Solving Some Linear and Non-Linear Parabolic Differential Equation. Journal of Applied Mathematics and Physics,03,1506-1521. doi: 10.4236/jamp.2015.311176