A Method for Solving the Three-dimensional Wave Equation

On the basis of the Laplace integral transform, locally one-dimensional scheme of cleavage and quasi-linearization method to obtain an approximate analytical solution of the three-dimensional nonlinear hyperbolic equation of second order. The assessment of the accuracy of analytical formulas when compared with the exact solution of the first boundary value problem and numerical solution by a known method.


Introduction
In mathematical modeling of heat and mass transfer [1], the heat transfer in high-frequency processes [2], vibrations [3] and so on. there is a problem the solution of telegraph type an equation [1][2][3]. If the solution of a nonlinear parabolic equation [1] there are a number of analytical techniques (reviewed in [4]), the exact analytical solutions are obtained for the linear one-dimensional (in the absence of a source) [5,6] or multi-dimensional [3] an equation of the telegraph type. However, in practice most often of interest to the solution of nonlinear boundary-value problems [1][2][3][4]7].
For one-dimensional solutions of nonlinear ordinary differential equations in [8] proposed a method of quasi-linearization. With this method, a decision along the nonlinear problem is reduced to solving a sequence of linear problems, which is essentially a development of the well-known Newton's method and its generalized variant proposed by L. V. Kantorovich [9]. Otherwise, the quasi-linearization − is the application of a nonlinear functional generated by the nonlinear boundary value problem, the Newton- Kantorovich. In the numerical solution of problems of mathematical physics were effective splitting methods [10,11]. In particular, the locally one-dimensional scheme cleavage [10] proposed to solve the multi-dimensional heat equation in combination with the analytical (constant coefficients) and numerical methods The purpose of article − with the help of the locally one-dimensional scheme splitting [11], quasi-linearization [8,9] and the Laplace integral transform [12] to find an approximate analytical solution of the nonlinear three-dimensional hyperbolic heat conduction equation in a finite region and to assess the accuracy of analytical formulas.

Statement of the Problem and the Algorithm of Method
Suppose you want to find a solution to a hyperbolic equation of the second order [1,3], with sources in the parallelepiped x x x x = (2) and to simplify further calculations with the boundary condition of the first kind , , whose value for the times considered below: We will always assume: , which continuously in t Q and has continuous derivatives Performed following conditions: , j = 1, 2, 3 in the general case can be non-linearly dependent on the solution of problem [1]. 3. View ) (T A is defined by the following formula (16); Ψ − given continuous function on the boundary Г for 0 0 t t ≤ < , having bounded partial derivatives of the first order. Applicable the locally one-dimensional scheme splitting of the equations (1) − (3) on the differential level [11] and introduce the superscripts (1), (2), (3) to denote the solution of the intermediate stages, as well as ξ there is the direction of wave decision and η there is direction the solution of the parabolic part equation (1). Then we have ), , ), , ) , , ( where ξ + η = 1, We are talking about the next model, for example, the conductive-convective heat transfer for the hyperbolic heat   , According to this model, the process the conductive-convective heat transfer "stretched" in time and takes place during the time of the gap 3 * t [10], and instead * t . Such an approach to solve multi-dimensional equations of partial differential equations with constant coefficients is proposed and justified in [10,11]. For the wave equation But before that, to the system (4) -(15) must apply Kirchhoff transformation [4] and quasi-linearization [8,9] to obtain the differential equations with constant coefficients, which can be solved with Laplace integral transform [12].
In the future, to use the inversion formula We use Kirchhoff transformation [4] dT A Then, taking into account the relations [4]: (24) Then the final solution of the problem (1) (28) Note that the range of the independent variables and the type of boundary conditions do not change under the transformation Kirchhoff (17), and in the presence of the inversion formula boundary conditions of the first kind of goes to the Dirichlet condition.
Our purpose to receive a solution of the nonlinear boundary value problem, if it exists, as the limit of a sequence of solutions of linear boundary value problems. To do this, we use the results [7][8][9]. Assume further that all the coordinate directions in space are equivalent.
) , , ( where y there is the any coordinate from j Then in a quasi-one dimension of equation (29), (30) can be rewritten in the coordinate 1 Expressions similar to (29), (30) can be written in other coordinate directions 3 2 , x x . In particular, the second coordinate direction 2 x is necessary in (31), (32) everywhere to replace the top and bottom indexes (1) and 1 on (2) and 2, and the top index (0) on (1). Thus for the entry condition in the second co-ordinate direction Each function The algorithm follows from the approximation method of Newton-Kantorovich [9] in the functional space.
In order to reduce further records introduce the following notation: A Method for Solving the Three-dimensional Wave Equation Obtain a quasi-one dimension solution of the problem (31), (32) in coordinate direction 1 x , using equation (20) − (22) Laplace integral transform is applicable to the differential equation (37) Using the inverse Laplace integral transform [12]: [12] ) , The derivative 1 1 / x g ∂ ∂ in (38) we find, using the second boundary condition of (36) , Transform the expression on the right-hand side of (40) so as to get rid of the integral with variable upper limit. Then, by introducing the Green's function ) , ( 1 1 y x E [7,8] (40) is rewritten in the return of the upper index (1) and lower ξ, noting that b, A is clearly not dependent on We apply again Kirchhoff transformation (17) to return to the original variable T in equation (4) Since the second term on the left-hand side of equation (42) takes the form , then it is necessary to reapply the quasi-linearization (31), (32), then we have We use the idea of Doetsch [13] on the applicability of Laplace integral transform to the partial differential equation as many times as its dimension. Then, with the initial conditions (35), we obtain [7,12] of (44) ) , Now we obtain an analytical solution of the problem of the intermediate (23), (24), using the notation (33) Using the inverse Laplace integral transform [12]: , to restore the original for ) , ( x t v of (49) [12] ) , We find the derivative , Transform the expression on the right-hand side of (52) so as to get rid of the integral with variable upper limit. Then, by introducing the Green's function ) , ( 1 1 y x G [7,8] ) , expression (51), using the formulas (33) and the return of a subscript * , η, as well as the top − (1), is rewritten As a result, the solution of (54) takes the form [14]: is taken from the formula (45) using equation (19).
Using the results of [7,15], we find the condition for the unique solvability of the problem (1) − (3) under certain assumptions, and get a quadratic rate of convergence of the iterative process.

Test Results of Inspections
The accuracy of the analytical formulas (19), (27), (45) − (47), (55) -(61) check on the test function in the solution of boundary value problems for partial differential equations in A Method for Solving the Three-dimensional Wave Equation

126
It was taken the exact solution (62) - (64) ) exp( then the source of F in (62) has the form The following reference values were used input: m = 0.5, w = 1, ξ = η = 0.5, The program is in G-Fortran, the calculation was performed on PC Pentium (3.5 GHz) with the double accuracy. In table 1 gives the maximum relative error in percent  [18]. For the numerical computation applied non-explicit unconditionally stable difference scheme with an absolute error of approximation for the first and second

Conclusion
1. On the basis of the locally one-dimensional scheme splitting, quasi-linearization and Laplace integral transform find an approximate analytical solution of a nonlinear hyperbolic equation of the second order, without using the theory of the series [17]. 2. In the one case, a comparison of the accuracy of analytical formulas articles with known exact solution of the telegraph type [5]