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NUMERICAL METHOD FOR SOLVING SYSTEMS OF

CONSERVATION LAWS OF MIXED TYPE

By: Nid`a Rababah

Advisor: Prof. Kamel Al-Khaled

04.05.2020.

Introduction

Introduction

01.

Nonlinear partial differential equations.

02.

Conservation laws.

03.

Adomian decomposition method

04.

Motivation

Nonlinear PDE

  • Nonlinear PDEs are very important in a variety of scientific fields, especially in applied mathematics, engineering, physics, solid-state physics, plasma physics, plasma waves, capillary-gravity waves, and chemical physics.

  • The availability of exact solutions for nonlinear equations can greatly facilitate the verification of numerical solvers on the stability analysis of the solution.

Conservation laws.

  • Conservation laws are a common feature of individual theories of continuum physics.

  • The laws are supplemented by constitutive relations which characterize the particular medium in question by relating the values of the main vector field U to the flux F.

  • The conservation laws led to the system of partial differential equations of the first order which can be written as:

which can be written equivalently in the following form

ADM

  • The Adomian decomposition method (ADM) was developed from the 1970s to the 1990s by George Adomian.

  • Over the last thirty years the Adomian decomposition approach has been applied to obtain formal solutions to a wide class of both deterministic, and stochastic PDEs

  • In recent years, the decomposition method has emerged as an alternative method for solving a wide range of mathematical problems.

  • The main idea of ADM is to find a solution in the form of a series which can be determined by the recurisve relationship.

  • ADM is very efficient, convergent, and can be applied to a large class of problems.

Motivation

  • We illustrate in detail all the steps required for the construction of all possible solutions by using ADM to each proposed model.

  • Graphical analysis is provided to recognize the physical shapes of all obtained numerical solutions.

  • Tables are provided to illustrate the accuracy of the error of some obtained solutions.

  • Finally, all the required computational work is performed by using Mathematica software.

ADM

ADM

  • Structure of ADM

  • Advantages

Structure of ADM

1) We consider the system of nonlinear PDEs in the operator form:

with initial conditions:

2) Writing the unknown linear functions and the nonlinear operators in terms of an infinite series of components

3)Applying the inverse operator to the system and using the given conditions we obtain:.

* Adomian polynomials can be constructed by using the general formula

4) Using the recursive relation:

5) Taking the limit to find the exact solutions for u(x,t) and v(x,t).

Advantages

1) The main advantage of the method is that it reduces the size of computation work and maintains the high accuracy of the analytical solution in terms of rapidly convergent series.

2) It provided a direct scheme that can be solved by recursive relationship by using a few iterations for both linear, nonlinear deterministic and stochastic equations, without the need for linearization, perturbation, massive computation and any transformation.

3) It‘s very efficient for nonlinear models, and it`s results give evidence that high accuracy can be achieved.

Solving System

Solving System

The models solved by the ADM are:

1) The shock wave equation

2) The P-system

3)The Cyclic system

shock wave equation

  • Shock waves are very small regions in the gas where the gas properties change by a large amount.

  • The shock wave equation describes the flow of gases.

The analytic solution of the shock wave equation which describes the flow of most gases as given by [5,6,7]

Solution

The analytic solution

- Based on the ADM and the recurrence relationship:

The approximate solution

Graphical analysis

The exact solution

P-system

  • P-system is describing the one-dimensional isothermal motion of a compressible elastic fluid-solid can be described in Lagrangian coordinates by the nonlinear coupled system which describes the one-dimensional longitudinal isothermal motion in elastic bars, or fluids.

The analytic solution

- We choose a problem with known solution allows for more complete error analysis so to derive an approximate solution for the p- systemwe start with the one-dimensional conservation law

such that:

with the approximation to the Riemann type condition:

- The transformation :

- Now we get a new system given by

- To solve this system we consider the coupled system of nonlinear PDE of the form:

by using the transformation

- By using of aforementioned techniques of ADM we reach the following steps as follow:

1) Applying the inverse operator to find the terms of the decomposition series:

2) Representing the unknown functions as infinite series:

3) Finding the values of H(x,t) by using the initial conditions:

4) Using the recursive relationship:

Graphics and Tables

The approximate solution of u(x,t)

The exact solution of u(x,t)

The approximate solution of v(x,t)

The exact solution of v(x,t)

  • Tables (4.1,4,2) indicate that the results of the suggested method are getting very close to the exact solution even when a small number of terms are used.

  • The error can be made smaller by adding more terms to the decomposition series

Cyclic system

The problem

has a local solution in the interval (0,T), where T is some small constant.

- It might be probable that the scheme will diverge for that T.

- To run the program in the interval (0,T) we find a smaller time interval in which scheme will converge, and solve the system using the given conditions.

Continuing in this way

solve the system

smaller time

the system has a solution

References

KEEP

IN

MIND

KEEP IN

MIND

1. M. J. Ablowitz, P. A. Clarkson; Nonlinear evolution equations and inverse Scattering, Cambridge University Press, (1991).

2. Fan, Hai Tao; A vanishing viscosity approach on the dynamics of plase transitions in Van der Waals fluids. J. Differential Equations, Volume 103,pp. 527-39 (1993).

3. Peter D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock, SIAM, NY (1973).

4. Chi-Wang Shu; A numerical method for systems of conservation laws of mixed type admitting hyperbolic flux splitting Journal of Computational Physics, Volume 100, Issue 2, June (1992), pp. 424-429.

5. K. Al-Khaled, Theory and Computations in Hyperbolic Model Problems, Ph.D. Thesis, University of Nebraska-Lincoln, USA, 1996

6. C.Y. Chow ,An Introduction to Computational Fluid Mechanics Wiley, N.Y. (1979)

7. J. Kevorkian,Partial Differential Equations, Analytical Solution Techniques ,Wadsworth and Brooks, N.Y. (1990)

8. E.D. Banta Lossless propagation of one-dimensional, finite amplitude sound waves J.Math. Anal. Appl., 10 (1965), pp. 166-173

9. Adomian, G., 1988. A Review of The Decomposition 29. Duan, J.S., 2010. Recurrence Triangle for Adomian Method in Applied Mathematics, J. Math. Anal. Polynomials, Appl. Math. Comput., 216: 1235-1241. Appl., 135: 501-544.

10. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston (1994).

11. Mustafa Inc, ON NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS BY THE DECOMPOSITION METHOD, Department of Mathematics, Firat University, 23119 Elazig Turkiye, Kragujevac J. Math. 26 (2004) 153–164.

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15. S. Guellal, P. Grimalt, Y. Cherruault, Numerical study of Lorentz’s equation by the Adomian method, Comput. Math. Appl., 33 (3) (1997), 25–29.

16. Rachana Khandelwal, Padama Kumawat and Yogesh Khandelwal, Kamal decomposition method and its application in solving coupled system of nonlinear PDE’s, Department of Mathematics, Maharishi Arvind University, Jaipur, Rajasthan – 609 307, India, Malaya Journal of Matematik, Vol. 6, No. 3, 619-625, 2018.

17. Y. Cherruault, Convergence of Adomian’s method, Kybernetes, 18 (1989), 31–38.

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19. FathiMAllan, Kamel Al-Khaled, An approximation of the analytic solution of the shock wave equation, Journal of Computational and Applied Mathematics, Volume 192, Issue 2, 1 August 2006, Pages 301-309.

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21. Marwan T. Alquran, Kamel M. Al-Khaled, Numerical Comparison of Methods for Solving Systems of Conservation Laws of Mixed Type, Vol. 5, 2011, no. 1, 35 - 47.

22. Al-Khaled K, A Sinc-Galerkin approach to the p-system, Vol. 16(1998), 1-13.

23. N. Ngarhasta, B. Some, K. Abbaoui, Y. Cherruault, New numerical study of Adomianmethod applied to a diffusion model, Kybernetes 31 (2002) 61–75.

24. G. Adomian, Solving Frontier Problems of Physics: The decomposition method, Kluwer Acad. Publ., Boston, 1994.

Thank you

The End

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