A Reliable Numeric-Analytic Treatment for the p-System
Abstract. A residual power series scheme is developed for mixed-type p-system of conservation laws. The residual power series method (RPSM) does not require discretization, reduces the system to an explicit system of algebraic equations and consequently of massive computations. Convergence hypotheses are discussed and error bounds of exponential rates are derived. The scheme is numerically tested on the van der Waals equations in fluid dynamics. Stability is obtained graphically while transition through regions. The reported results reveal the reliability, efficiency and economical implementation of the proposed algorithm for handling nonlinear partial differential equations in applied mathematics.
A first order partial differential equation of the form
is known as a scalar conservation law in one space dimension. represents the conserved quantity, is the flux, is the one-dimensional space variable, while denotes the time. This type of equations often describes transport phenomena.
Laws of conservation state that certain measurable quantities do not change during time and within an isolated physical system. In classical physics and mechanics, laws of this type govern energy, momentum, angular momentum, mass, and electric charge. In particle physics, other conservation laws apply to properties of subatomic particles that are invariant during interactions.
The purpose of this contribution is to construct highly-accurate, stable, semi-analytic solution for mixed-type hyperbolic-elliptic systems of conservation laws originated in models of one-dimensional isothermal motion for a compressible elastic fluid. The mixed type p-system
with Riemann-type initial data
where is the velocity, is the specific volume and is the pressure. Recall that, such system is hyperbolic at a point on a given domain if the Jacobian of the flux vector has real eigenvalues. Moreover, if these eigenvalues are distinct, the system is strictly hyperbolic. In the case of complex conjugate eigenvalues, is called elliptic at that point. The set of all points at which the system is elliptic is called elliptic region. Systems with nonempty two regions, hyperbolic and elliptic are called mixed systems. As the pressure in Eq.(2) varies for varied materials. Typically, for an ideal gas, is strictly increasing, so the system is hyperbolic. However, for some material models may not be monotone which implies mixed-type system.
In general, the well-posedness theory of mixed systems did not develop yet such as in the case of hyperbolic systems 1. The ill-posedness of the system in the elliptic region reveals the possibility of shocks. That is, the state in the elliptic region is not stable, and it typically evolves into phase transitions. Many works discussed mixed systems and their applications theoretically 2-6. A collection of the numeric-analytic attempts on solving mixed systems have been carried out in 6-11 and references therein.
The structure of this paper is organized as follows: Methodology and basic concepts of the residual power series method (RPSM) is improved to tackle coupled of balance laws (conservation laws with source terms) in section 2. Analytic Application of this algorithm for solving p-system in Eq.(2) is given in section 3. For further clarification and accurate purposes, the one-dimensional unsteady flow of a van der Waals gas, whose constitutive function is given by
where and are all positive constants, is considered. Discussions and conclusions are given in section 5.
2. METHODOLOGY AND ERROR ANALYSIS
The RPSM was first proposed by Abu Arqub 12 for the treatment of linear and nonlinear ordinary differential equations of integer and fractional orders. A wide range of applications to fractional partial differential equations have been executed, see 13-17 and references therein.
Consider the generalized one-dimensional system of balance laws, that describes many physical phenomena, of the form
where , , and are assumed to be analytic on . In more compact form, and using the vector symbols, system in Eq.(5) can be written as
In this section, we illustrate the methodology of the generalized residual power series technique for treating the system in Eq.(5). The RPSM assumes the solutions in the form of power series
For , define the -order approximate solutions, that satisfy initial conditions, by the truncated series
The first approximate solution, with in Eq.(8), of would be
is the analytic -residual vector function. Continuing this process for , the unknown coefficients could be determined consecutively by solving the algebraic equations
Accordingly, the following theorem has been proved:
Theorem 2.1. Suppose that the functions has the power series representation at in Eq.(7) on the domain , for some closed interval , is the radius of convergence for the power series. If , then the unknown coefficients, for , are given by
For if is analytic on the indicated domain, general solutions will be as in Eq.(7), provided that the series have closed forms.
Also, the following facts for the RPSM were proved 18:
Theorem 2.2. The residual function vanishes as approaches the infinity.
Theorem 2 tells that the residual function defined in Eq.(9) is infinitely many times differentiable at .
Corollary 2.3. The truncated series solutions given in Eq.(8) and obtained by the GRPSM are the – order Taylor expansion of respectively about .
Proof: It is obvious using Eq.(10).
As a result of Theorem 3, and because of convenience to have a notation on error analysis for the approximations using proposed technique. Taylor’s theorem implies that
for some , is the vector of reminder terms. Assume that , then positive constants , and exist with , and in the sense of norm. So, one can easily prove the following result:
Corollary 2.4. Suppose that . If the truncated series solutions given in Eq.(8) is used to approximate the solution vector function of the initial-value problem Eq.(6), then error bounds are estimated to be
The error bound for is something large since the values of are of distant from the center . For the numerical accuracy and comparison purposes of RPSM, absolute and relative errors, defined at each point , are given by
respectively. This is useful with given closed-form exact solution. In the case of unknown exact solution, absolute errors will be defined in two ways as following:
In other words, truncated semi-analytic solution , obtained using the proposed scheme, converges to exact solution , if a number exists so that , , and , if . is called the -convergence rate. Faster convergence can be obtained if as .
3. ANALYTIC SOLUTION OF THE P-SYSTEM
To illustrate the procedure discussed in the previous sections for solving the p-system Eq.(2) subject to initial date in Eq.(3), the RPSM is designed to deal with continuous initial data, so the following approximate transformations 8
where is a suitability constant, would be useful for implementation. Consequently, transformed initial conditions are
The unknown truncated-series coefficients Eq.(8) will be found consecutively subject to the associated residual functions, for
provided that the summation converges and has an exact closed-form. Taking into account Eq.(21), the first few terms of series solution for are listed to be
More and more series solution coefficients can be obtained with aid of Mathematica software package.
4. VAN DER WAALS SYSTEM – NUMERICAL ILLUSTRATIONS
This section aims to verify the convergence and stability of the proposed algorithm to van der Waals equations with pressure given in Eq.(4), , and 11. According to Eq.(18), the converted initial conditions will be
The Adomian decomposition method (ADM) 8, sinc-collocation method (SCM) 9, variational iteration method (VIM) 10, and reduced differential transform method (RDTM) 11 were developed to treat this problem. Because of the complexity of computing integrals, few number of iterations could be found using the ADM, VIM and RDTM, with more stability in the last case. The SCM is of less stability. The residual power series procedure is operated to solve the van der Waals equations, more and more terms of series solutions Eq.(7) could be determined since it is based on reducing the system of differential equations to system of algebraic equations. Approximate solutions are plotted in Figure 1. Our system is of mixed type, the initials start outside the elliptic region, then enter this region while and change back to hyperbolic while as shown in Figure 2. It can be concluded that the present method is more stable in comparison to others. Furthermore, with 5-terms approximate solutions, absolute errors given in Eq.(15) are listed in Table 1. The RPSM is of highly accurate that makes the obtained solutions acceptable as a criterion of comparison in next works.
the applications to the van der Waals gas equations, efficiency, simplicity and convenience of the generalized residual power series method were obtained in comparison to other existing methods. Analytic solutions in a form of high accurate truncated series solutions are derived. Numerical results reveal that the generalized scheme is stable, as shown in Figure 1 and Figure 6, while system changes the type. Also, It may be concluded that the RPSM is very powerful and efficient technique in finding analytic solutions for wide classes of problems and can be also easy to be extended to other non-linear diffusion equations, with the aid of Mathematica.
2. S. Berres, R. Burger and A. Kozakevicius, Numerical approximation of oscillatory solutions of hyperbolic-elliptic systems of conservation laws by multiresolution schemes, Adv. Appl. Math. Mech., 1 (5) (2009), pp. 581-614.
3. B.L. Keyfitz, Mathematical properties of nonhyperbolic models for incompressible two-phase flow. In: Proceedings of the International Conference on Multiphase Flow, New Orleans, May 27–June 1, 2001.
4. D. Marchesin and B. Plohr, Theory of three-phase flow applied to water-alternating-gas enhanced oil recovery. In: H. Frestuhler and G. Warnecke (EDS.), Hyperbolic Problems: Theory, Numerics, Applications, Vol. II, Birkhauser Verlag, Basel (2001), pp. 693–702.
5. A.D. Fitt, Mixed systems of conservation laws in industrial mathematical modelling, Surv. Math. Indust., 6 (1996), pp. 21–53.
6. H. Holden, L. Holden, N.H. Risebro, Some qualitative properties of 2×2 systems of conservation laws of mixed type, in: B.L. Keyfitz, M. Shearer (Eds.), Nonlinear Evolution Equation that Change Type, Springer-Verlag, 1990, pp. 67–78.
7. E.A. Az-Zo’bi and K. Al Khaled, A new convergence proof of the Adomian decomposition method for a mixed hyperbolic elliptic system of conservation laws, Appl. Math. Comput. 217 (2010), pp 4248–4256.
8. E.A. Az-Zo’bi, Construction of solutions for mixed hyperbolic elliptic Riemann initial value system of conservation laws, Appl. Math. Modeling, 37 (2013), pp 6018-6024.
9. Kamel Al-Khaled, Cardinal-type approximations for conservation laws of mixed type, NONLINEAR STUDIES, 21(3) (2014), pp. 423-433.
10. E.A. Az-Zo`bi, On the Convergence of Variational Iteration Method for Solving Systems of Conservation Laws, Trends in Applied Sciences Research, 10 (3) (2015), pp 157-165.
E.A. Az-Zo’bi, K. Al Dawoud and M. F. Marashdeh, Numeric-analytic solutions of mixed-type systems of balance laws, Applied Mathematics and Computation, 265 (2015), pp 133–143.
O. Abu Arqub, A. El-Ajou, A. Bataineh, I. Hashim, A representation of the exact solution of generalized Lane-Emden equations using a new analytical method, Abstract Appl. Anal., (2013) Article ID 378593, 10 pages, doi:10.1155/2013/378593.
Syam, M.I., Analytical Solution of the Fractional Initial Emden–Fowler Equation Using the Fractional Residual Power Series Method, Int. J. Appl. Comput. Math, 4 (2018): 106. https://doi.org/10.1007/s40819-018-0538-2
Z. Korpinar and M. Inc, Numerical simulations for fractional variation of (1?+?1)-dimensional Biswas-Milovic equation ,Optik,166 (2018), pp 77-85
I. Jaradat, M. Alquran and K. Al-Khaled, An analytical study of physical models with inherited temporal and spatial memory, The European Physical Journal Plus, April (2018), pp 133-162
A. Arafa and G. Elmahdy, Application of residual power series method to fractional coupled physical equations arising in fluids flow, International Journal of Differential Equations (2018), Article ID 7692849, 10 pages https://doi.org/10.1155/2018/7692849
S. Effati, S.A. Rakhshan and S.e Saqi, Formulation of Euler–Lagrange equations for multidelay fractional optimal control problems, J. Comput. Nonlinear Dynam., 13 (6), 061007 (May 02, 2018), 10 pages