Bernoulli Matrix Approach for Solving Two Dimensional Linear Hyperbolic Partial Differential Equations with Constant Coefficients

The purpose of this study is to give a Bernoulli polynomial approximat ion for thesolution of hyperbolic partial differential equations with three variables and constant coefficients. For this purpose, a Bernoulli matrix approach is introduced. This method is based on taking the truncated Bernoulli expansions of the functions in the partia l d ifferential equations. After replacing the approximations of functions in the basic equation, we deal with a linear algebraic equation. Hence, the result matrix equation can be solved and the unknown Bernoulli coefficients can be found approximately. The efficiency of the proposed approach is demonstrated with one exampl e.


Introduction
When a mathematical model is formu lated for a physical problem, it is often represented by partial differential equations, that are not solvable exact ly by analytic techniques. Therefore, onemust resort to approximation and numerical methods. For examp le,theTelegraph equation (as a hyperbolic partial d ifferential equation) is co mmonly 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 bio logy, 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 mathemat icians and physicists.The various methods for solving linear hyperbolic partial d ifferential equations are introducedin [4].
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 with the in itial conditions 1 2 ( , ,0) ( , ), ( , ,0) ( , ) u u x y x y x y x y t ψ ψ ∂ = = ∂ (2) and the boundary conditions . The considered partial differential equation (1) arise in connection with various physical and geometrical problems in which the functions involved depend on two or mo reindependent variables, on time t and on one or several space variables [4]. For

Fundamental Relations
To obtain the numerical solution of the hyperbolic part ial 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 ( , , ) identity mat rices respectively.
By using the relations (6) Substituting the expressions (5)- (8) and (10) into equation(1) and simplifying the result, we have the matrix equation Briefly, we can write equation (11) in the form WA=F, (12) whereÎ is the 3 3 ( We now present the alternative forms for ( , , ) u x y t which are important for simp lifying matrix forms of the conditions. The simp lification in conditions is done only with respect to Differential Equations with Constant Coefficients the variables , x y and t . Therefore we must use different forms for init ial and boundary conditions. For the init ial conditions (2), Notice also that the matrices involved in the right-hand side of Eqs. (2) and (3) (2) and (3) and then simplify ing the result, we get the matrix forms of the conditions, respectively, as 1 1 1 ; The unknown Bernoulli coefficients are obtained as ( )

1ˆ,
W G is generated by using the Gauss elimination method and then removing zero rows of the Gauss eliminated matrix. Here W and G are obtained by throwing away the maximu m nu mber of row vectors fro m W and G, so the rank of the system defined in (28) cannot be smaller than 3 ( 1) N + . This process provides higher accuracy because of the decreasing truncation error.

Accuracy of the Solution and Error Analysis
We can easily check the accuracy of the method. Since the truncated Bernoulli series (4)