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In this paper, two different numerical schemes, namely the Runge-Kutta fourth order method and the implicit Euler method with perturbation method of the second degree, are applied to solve the nonlinear thermal wave in one and two dimensions using the differential quadrature method. The aim of this paper is to make comparison between previous numerical schemes and detect which is more efficient and more accurate by comparing the obtained results with the available analytical ones and computing the computational time.

Thermal wave is reaction-diffusion equation that plays an ever-increasing role in the study of material parameters. It has been employed in optical investigations of solids, liquids and gases with photo-acoustic and thermal lens spectroscopy. Thermal waves have also been used to analyze the thermal and thermodynamic properties of materials and image thermal and material features within a solid sample [

In the past several decades, there has been greeting activity in developing numerical and analytical methods for the thermal wave equation. Due to the nonlinearity and complexity of such problems, only limited cases can be analytically solved [2-5]. Yan applied the projective Riccati equation method to solve Schrodinger equation in nonlinear optical fibers [

In this research, the thermal wave propagation model is solved by using two numerical methods to make comparison between them. In the first method, we used the hybrid technique method of Runge-Kutta fourth order method (RK4) and differential quadrature method (DQM). In the second method, we used the combined algorithm of DQM, Perturbation method of second degree and implicit Euler method. Perturbation method is used to avoid the nonlinear term. The obtained results are compared with the previous analytical ones to complete the comparison between previous different numerical schemes.

Propagation of thermal waves through a rectangular plate, is governed by [

where: U is a temperatureα and β are diffusion parameters in direction of x and y, respectivelyγ is reaction parametera and b are plate dimensions in direction of x and y, respectivelyU_{max} is a maximum temperature of the system.

Along the external boundaries, the temperatures can be described as:

where

Then initial temperature may be described as:

where

The main strategy is to employ DQM to reduce the problem to a system of ordinary differential equations then to apply RK4 to solve the reduced system as follows:

1) Discretize the spatial domain using Chebyshev-Gauss-Lobatto grid points [

2) Apply the method of differential quadrature in terms of nodal temperature, such that:

where

3) On sustainable substitution from Equations (5) into (1), one can reduce the problem to the following system of ordinary differential equations as:

or simply

where

4) Update the temperature using RK4 such that [

where

where

The main strategy is to apply perturbation method of second order [17,18] then applying DQ discretization to reduce the problem to a system of ordinary differential equations then applying implicit Euler method to transform the previous system to a system of linear algebraic equations as follows:

1) We can solve

subjected to the prescribed to boundary and initial conditions in Equations (2) and (3), assuming

where

The following condition is tested to ensure the convergence condition [

2) On sustainable substitution from Equation (11) into (10), one can reduce the problem to the following equation.

3) Applying zero order perturbation method such that,

Subjected to boundary and initial conditions in Equations (2) and (3), where differential quadrature method and implicit method are used to reduce Equation (14) to a system of linear algebraic equations such thatBy substitution of Equation (5) into (14) result that,

4) First order perturbation method is applied such that,

Subjected to the same boundary and initial conditions in Equations (2) and (3), reduced to the following algebraic system in equations

5) Also second order perturbation method is applied such that,

Subjected to the same boundary and initial conditions in Equations (2) and (3), reduced to the following algebraic system

Finally, the series solution can be written as

We carry on previous procedure until the specified time is reached.

To ensure the accuracy of the proposed numerical techniques, the thermal wave propagating model is solved using presented methods and compared with the available analytical solution [14,20].

Consider a one-dimensional problem of thermal wave propagation along x-direction as:

While

The exact solution for such problem can be obtained as [

To validate the accuracy of numerical results, the following errors [

For the numerical computation, the time domain is limited to

In the obtained results the advantage of using an implicit scheme has been observed. Stability problems are not encountered due to the use of implicit time integration step and larger time increments can be used, e.g. for t = 30,

Consider also a simple two dimensional problem with

To show the effect of oscillation on figure we graph the absolute error in range 10 ≤ t ≤ 11

which can be solved exactly as [

Throughout this study, thermal wave propagation model which is the type of reaction-diffusion equations is solved by using DQM for space discretization and two different time-integration schemes. Moreover, one can use a small number of discretization points, which lead to higher accuracy. Also for the nonlinear wave equation, the use of DQM with non-uniform grid discretization increases the accuracy and stability of solution. The resulting system of ordinary differential equations is solved by using two different time integration schemes in order

to make comparison between two methods and detect which of them is better. The numerical results obtained in this paper ensure that the problems have small desired time to reach it. Thus they have very small step size which is preferred and use RK4 to solve the system of ordinary differential equations in order to decrease the computational time. On the other hand, the problems which have high desired time to reach it, thus have large incremental time (stiff problems) which are preferred and use implicit Euler with perturbation method to solve the system of ordinary differential equations.