The Galerkin-Implicit Method for Solving Nonlinear Variable Coefficients Hyperbolic Boundary Value Problem

This paper has the interest of finding the approximate solution (APPS) of a nonlinear variable coefficients hyperbolic boundary value problem (NOLVCHBVP). The given boundary value problem is written in its discrete weak form (WEFM) and proved have a unique solution, which is obtained via the mixed Galerkin finite element with implicit method that reduces the problem to solve the Galerkin nonlinear algebraic system (GNAS). In this part, the predictor and the corrector techniques (PT and CT, respectively) are proved at first convergence and then are used to transform the obtained GNAS to a linear GLAS . Then the GLAS is solved using the Cholesky method (ChMe). The stability and the convergence of the method are studied. Some illustrative examples are used, where the results are given by figures that show the efficiency and accuracy for the method.


1.Introduction
Hyperbolic partial differential equations arise in many physical problems, such as vibrating strings, and in many other fields such as fluid dynamics, optics, and others. In general, there are many researchers who are interested in the solution of boundary value problems, in ISSN: 0067-2904 particular the solution of the nonlinear hyperbolic boundary value problem (NLHBVP). In 2015, Feller used the Lévy Laplacian to solve a NOLVCHBVP [1]. In 2017, Mardani et al. used the Moving Least Squares method for the nonlinear hyperbolic telegraph equation with variable coefficients [2]. Ashyralyev and Agirseven, in 2018, solved a NOLHBVP with a time delay [3]. While in 2018, Ahmedatt et a.l looked at some nonlinear hyperbolic -Laplacian equations [4]. Adewole, in 2019, found the APPS of a linear hyperbolic (LHBVP) [5].
The finite element method has been studied by many researchers who are interested in this field to solve LHBVP. For example, in 2014, Quarteroni studied in his book the numerical solution for LHBVP and some especial types NOLHBVP by using GFEME [6]. In 2018, Wick studied in his book the GFEME for solving LHBVP and NOLHBVP with constant coefficients [7]. In this paper, we care about the study of the APPS of the NOLVCHBVP. The given boundary value problem is written in its WEFM, and then it is discretized using the mixed Galerkin finite element method (GFEME) for the space variable with the implicit method (IM) for the time variable (MGFEIM). It is proved that the discrete problem has a unique solution. The problem then reduces for solving the GNAS. In this point, the PT and CT are used to transform the GNAS to a GLAS, which is solved by using the ChMe. The stability and the convergence of the method are studied. A computer program is codding in Matlap to find the APPS for the problem. Some illustrative examples are given and the results are given by figures, which show the efficiency and accuracy for the considered method .

Assumptions (i)
Let κ 1 and κ 2 be two positive constants such that the following are satisfied: The function is defined on × , continuous with respect to which satisfies the following: where is a Lipchitz constant and .

Discretization of The Continuity Equation (COE)
By setting in the WEFM of (5-7), then it is discretized by using the GFEME as follows: let the domain is divided into sub regions , let { } be a triangulation of ̅ , and let { } be a subdivision of the interval ̅ into (n) intervals, where [ ] of equal length ∆ = . Also, let ) be the space of continuous piecewise affine functions in . The discrete equations (DES) are written as follows:

The APPS of the NOLVCHBVP
To find the APPS ̅ for the DES (8)-(11), using the MGFEIM, the following steps are used: 1) Let { ⃗ } be a finite basis of , with using the GFEME, let ) be an APPS of (8-11), then one has where and ( ) for each are unknown constants. 2) Using the APPs in (8-11) to get : System (12)-(15) is GNAS and has a unique solution. To solve it, first we solve the GLAS (14) and (15) to obtain and , then the PT and the CT are utilized to solve (12) for each ( ) as follows: In the PT, we suppose that in the components of ⃗⃗ in the R.H.S of (12), then it turns to a GLAS, solving this system to get the predictor solution . Then, in the CT, we resolve (12) with setting ̅ (in the components of ⃗⃗ of the R.H.S of it) to get the corrector solution . Then we substitute in (13) to get . We can repeat this procedure if we want more than one time; this repetition can be expressed as follows: Equation (17)  (29) Also, by using the same way into the inequality (28), we get is ST in (30) By using theorem 3.2 in [9], there are subsequences of ({ }, { }, { } and of { }, { }, { },). Using the same notations again, they converge weakly to some in , to some in , which means that and for each , then That is, the limit point is a solution to the WEFM in the COE.

Cholesky Factorization
The Cholesky decomposition is used to solve the GLAS with two conditions, in which the coefficient matrix must be a symmetric and positive definite. Then the matrix can be factorized into the product of an Upper triangular matrix and Lower triangular matrix [8], and can be determined as shown in the following steps: Step 1: ∑ Step 2: ( ∑ ) for .

Numerical Examples
The problems in the following examples are coded by Matlap software: Example 1: Consider the following NOLVCHBVP: with the variables coefficients are: The exact solution (EXS) of this problem is ⃗ √ . Using the MGFEIM to solve this problem for , and , then the results are shown in Figure 1. (a) which shows the APPS, and Figure 1.(b) which shows the EXS at ̂ 0.5

Numerical Discussion and Conclusions
The MGFEIM is used successfully to solve the discrete of the WEFM of a certain type of NOLVCHBVP. The existence theorem of a unique convergent APPs is proved. The convergence of the PT and CT, which are used to solve the GNAS that is obtained from applying the MGFEIM, is proved and the ChMe, which is used inside these technique, is highly efficient for solving large GAS. The discrete of the WEFM proved that it is stable and convergent. The results of the considered examples showed the efficiency and accuracy of the method.