The Galerkin-Implicit Methods for Solving Nonlinear Hyperbolic Boundary Value Problem

This paper is concerned with finding the approximation solution (APPS) of a certain type of nonlinear hyperbolic boundary value problem (NOLHYBVP). The given BVP is written in its discrete (DI) weak form (WEF), and is proved that it has a unique APPS, which is obtained via the mixed Galerkin finite element method (GFE) with implicit method (MGFEIM) that reduces the problem to solve the Galerkin nonlinear algebraic system (GNAS). In this part, the predictor and the corrector technique (PT and CT) are proved convergent and are used to transform the obtained GNAS to linear (GLAS), then the GLAS is solved using the Cholesky method (ChMe). The stability and the convergence of the method are studied. The results are given by figures and shown the efficiency and accuracy for the method.


Introduction
Hyperbolic partial differential equations play a very important role as real life problems in many fields of sciences as in technology, fluid dynamics, optics, science and many others.
In the past few decades, there have been many researchers interested in their study to solve boundary value problems in general and in particular NLHBVE. Many researchers have used different methods to solve the NLHBVE, Smiley studied in 1987, was used Eigen function methods to solve problems of nonlinear hyperbolic value at resonance [1]. In 1989, Chi, Wiener, and Shah used in the exponential growth of solutions of nonlinear hyperbolic equations [2], while in 2001 Minamoto used the existence and demonstration of the uniqueness of solutions [3]. In 2004, Krylovas, and Čiegis, used the numerical asymptotic averaging for weakly nonlinear hyperbolic waves [4]. In 2018, Ashyralyev and Agirseven solved NLHBVE with a time delay [5].
The specific element method has been studied by several researchers interested in this field, for example, in 2010 Bangerth and Rannacher touched on Galerkin's specific adaptation techniques for wave equation [6]. Whereas, in 2017, Al-Haq and Muhammad discussed numerical methods to solve LHYBVP by difference method and the method of the specified elements [7].
In this paper, we are concerned the study of the APPS of the NOLHYBVP. The given BVP is written in its WEF, and in its discrete equation (DI) type. It is proved to have unique APPS. The APPS is obtained via the MGFEIM. The problem then reduces to solve the GNAS, then the PT and CT are proved convergent and are used to transform the GNAS to a GLAS. This GLAS is solved by using the ChMe. The stability and the convergence of the method are studied. A computer program is codding to find the numerical solution for the problem. The results are given by figures, and are shown the efficiency and accuracy for the method which is highly considered in this work.
(2) Using the APPs in (8)(9)(10)(11) to get , ∀ = 0,1, … , − 1: (3) System (12-15), is GNAS and has a unique solution. To solve it, we find at first 0 and 0 from solving (14) and (15) respectively, then, the PT and the CT are utilized to solve (12) for each ( = 0,1, … , − 1) as follows: In the PT we suppose +1 = in the components of ⃗⃗ in the R.H.S of (12), then it turn to a GLAS, which is solved to get the predictor solution +1 , then in the CT we resolve (12) with setting ̅ +1 = +1 (in the components of ⃗⃗ of the R.H.S of it) to get the corrector solution +1 , finally substituting +1 in (13) to get +1 , we can repeat this procedure if we want more than one time. This reputation can be expressed as follows: Equation (17) tells us the iterative method depending on just +1 ( +1) . Thus, equation (16) is reformulated as ( +1) = ( ( +1) ) , where is the number of the iterations. And this led us to the following theorem. Theorem (5): For any fixed point, the DES (8)-(11), and for Δ sufficiently small, has a unique solution = ( 0 , 1 , … . . , ) and the sequence of the corrector solutions converges on ℝ.
We back to (27) substituting = , the 1 st and the 3 rd term in the L.H.S are positives, then we use the above results in the R.H.S. of it , keeping in mind the first three terms in this side that are bounded (from the above steps), to obtain

Cholesky factorization
Cholesky method is used using to solve GLAS with conditions that the coefficient matrix must be a symmetric and positive definite. In this method the matrix can be factorized into the product of an Upper triangular matrix and Lower triangular matrix [11], and calculates as follows:

Conclusions
The MGFEIM is used successfully to solve the DI of the WEF of a certain type of NOLHYBVP. The existence theorem of a unique convergent APP is proved. The convergent 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 DI of the WEF is proved itis stable and convergent. The results are given by figures and show the efficiency and accuracy for the method.