1. Introduction

When a mathematical model is formulated for a physical problem, it is often represented by partial differential equations, that are not solvable exactly by analytic techniques. Therefore, onemust resort to approximation and numerical methods. For example,theTelegraph equation (as a hyperbolic partial differential equation) is commonly used in signal analysis for transmission and propagation of electrical signals and also has applications in other fields (see[1] and the references therein). We also note that,the differential equationsaresometimes linear in real world as is shown in papers[2, 3].

The solution of linear hyperbolic partial differential equations clarifies the linear phenomena which occur in many systems like as biology, engineering, aerospace, industry etc. In the recent years, noticeable progress has been made in the construction of the numerical solutions for linear partial differential equations, which has long been a major concern for both mathematicians and physicists.The various methods for solving linear hyperbolic partial differential equations are introducedin[4].

Since the beginning of 1994 several matrix (such as Legendre, Chebyshev, Bessel, Lagurre, Hermite, Bernstein etc.) approachesfor solving linear differential, integral, integro- differential, difference, and integro-difference equations, havebeen presented in many papers by Sezer and Cowork ers[4–11].

In this study, and in the light of the above mentioned methods (and by generalizing the Bernoulli matrix method for one dimensional PDEs[12]) we propose a new matrix approach which is based on Bernoulli truncated series in three dimension for solving two dimensional hyperbolic partial differential equations with constant coefficients in the following form

(1)

with the initial conditions

(2)

and the boundary conditions

(3)

and the solution is approximated by the following Bernoulli truncatedseries

(4)

so the Bernoulli coefficients to be determined are ; whereare known functions .

The considered partial differential equation (1) arise in connection with various physical and geometrical problems in which the functions involved depend on two or moreindependent variables, on time and on one or several space variables[4]. For and, equation(1) represents a two-space-dimensional damped wave equation (with a source term). The numerical solution of damped wave equations is of great importance in studying wave phenomena.

2. Fundamental Relations

To obtain the numerical solution of the hyperbolic partial differential equation with thepresented method, we first evaluate the Bernoulli coefficients of the unknown function. For convenience, the solution function (4) can be written in the matrix form

(5)

whereis the matrix defined as

and

where

On the other hand, the relations between the matrix and its derivatives are as follows:

(6)

(7)

(8)

where

whereand are and identity matrices respectively.

By using the relations (6)-(8)we have

We can also expand the function as a Bernoulli series:

(9)

From (9),one can obtain the following matrix form

(10)

where

with

.

Substituting the expressions (5)–(8) and (10) into equation(1) and simplifying the result, we have the matrix equation

(11)

Briefly, we can write equation(11) in the form

(12)

whereis the identity matrix and

We now present the alternative forms for which are important for simplifying matrix forms of the conditions. The simplification in conditions is done only with respect to the variables. Therefore we must use different forms for initial and boundary conditions. For the initial conditions (2),

(13)

while for the boundary conditions (3),

(14)

and also we use

(15)

,

Notice also that the matrices involved in the right-hand side of Eqs. (2) and (3) are of the form

(16)

(17)

(18)

(19)

(20)

(21)

By substituting relations (16)–(21) into (2) and (3) and then simplifying the result, we get the matrix forms of the conditions, respectively, as

(22)

(23)

(24)

(25)

(26)

(27)

(28)

The unknown Bernoulli coefficients are obtained as

Whereis generated by using the Gauss elimination method and then removing zero rows of the Gauss eliminated matrix. Here and are obtained by throwing away the maximum number of row vectors from W and G, so the rank of the system defined in (28) cannot be smaller than. This process provides higher accuracy because of the decreasing truncation error.

3. Accuracy of the Solution and Error Analysis

We can easily check the accuracy of the method. Since the truncated Bernoulli series (4) is an approximate solution ofEq. (1), when the function and its derivatives are substituted in Eq. (1), the resulting equation must be satisfiedapproximately; that is, for

and (positive integer). If max = (positive integer) is prescribed, then the truncation limit is increased until the difference at each of the points becomes smaller than the prescribed .

On the other hand, the error, can be estimated by and errors, and the root mean square error (RMS). We calculate the RMS error using the following formula :

where and are the exact and approximate solutions of the problem, respectively, and is an arbitrary time in

4. IllustrativeExample

This section is devoted to a computational result. We applied the method presented in this work and solved one example. We illustrate our work with the following example. The numerical computations were done using MATLAB v7.12.0.

Example.Consider the problem

.

The initial conditions are given by

By applying the technique described in the preceding section we find the matrix representation of the equation as follows:

Solving the above linear system yields to

which is the exact solution.

5. Conclusions

Two-space-dimensional linear hyperbolic equations with constant coefficients are usually difficult to solve analytically. For this purpose, the present method has been proposed for approximate solutions and also analytical solutions. It is observed that the method has a great advantage when the known functions in the equation can be expanded in Bernoulli series. Moreover, this method is applicable for the approximate solution of parabolic and elliptic type partial differential equations in two space dimensions. The method can also be extended to nonlinear partial differential equations, but some modifications are required.