2013
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A High Order Approximation of the Two Dimensional Acoustic Wave Equation with Discontinuous Coefficients
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2
This paper concerns with the modeling and construction of a fifth order method for two dimensional acoustic wave equation in heterogenous media. The method is based on a standard discretization of the problem on smooth regions and a nonstandard method for nonsmooth regions. The construction of the nonstandard method is based on the special treatment of the interface using suitable jump conditions. We derive the required linear systems for evaluation of the coefficients of such a nonstandard method. The given novel modeling provides an overall fifth order numerical model for two dimensional acoustic wave equation with discontinuous coefficients.
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1
15
Javad
Farzi
Javad
Farzi
Sahand University Of Technology
Sahand University Of Technology
I. R. Iran
Interface methods
two dimensional acoustic wave equation
high order methods
LaxWendroff method
WENO
discontinuous coefficients
Jump conditions
[[1] M. Dehghan and A. Mohebbi, High Order Implicit Collocation Method for the Solution of TwoDimensional Linear Hyperbolic Equation , Numerical Methods for Partial Differential Equations, Vol. 25, Issue 1, 232243, 2009. ##[2] J. Farzi and S. M. Hosseini, A High Order Method for the Solution of One Way Wave Equation in Heterogenous Media, Far East J. Appl. Math., Vol. 36, No. 3, 317330, 2009. ##[3] B. Gustafsson and P. Wahlund Time Compact High Order Difference Methods for Wave Propagation, 2D, . Sci. Comput., Vol. 25, pp. 195211, 2005. ##[4] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems., Cambridge University Press, Cambridge, 2004. ##[5] R. J. LeVeque, Wave Propagation Algorithms for Multidimensional Hyperbolic systems., J. comp. phys. 131,pp. 327353, 1997. ##[6] R. J. LeVeque and C. Zhang, The Immersed interface methods for wave equations with discontinuous coefficients., Wave Motion, 25, PP. 237263, 1997. ##[7] J. Qiu and C.W. Shu, Finite difference WENO schemes with LaxWendroff type time discretizations., SIAM J. Sci. Comput. 24, pp. 21852198, 2003. ##[8] C.W. Shu, Efficient Implementation of Weighted ENO Schemes., J. Comput. Phys., 126, pp. 202228, 1996. ##[9] C. Zhang and W.W. Symes, A Forth Order Method for Acoustic Waves in Heterogenous Media, Proceedings of International Conference on Mathematical and Numerical Aspects of Wave Propagation, 1998.##]
Chebyshev Spectral Collocation Method for Computing Numerical Solution of Telegraph Equation
2
2
In this paper, the Chebyshev spectral collocation method(CSCM) for onedimensional linear hyperbolic telegraph equation is presented. Chebyshev spectral collocation method have become very useful in providing highly accurate solutions to partial differential equations. A straightforward implementation of these methods involves the use of spectral differentiation matrices. Firstly, we transform telegraph equation to system of partial differential equations with initial condition. Using Chebyshev differentiation matrices yields a system of ordinary differential equations. Secondly, we apply fourth order RungeKutta formula for the numerical integration of the system of ODEs. Numerical results verified the high accuracy of the new method, and its competitive ability compared with other newly appeared methods.
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16
29
M.
Javidi
M.
Javidi
University of Tabriz
University of Tabriz
I. R. Iran
mo_javidi@yahoo.com
Chebyshev spectral collocation method
telegraph equation
numerical results
RungeKutta formula
[[1] A. Mohebbi, M. Dehaghan, High order compact solution of the one dimensional lin##ear hyperbolic equation, Numerical method for partial differential equations, 24 (2008)##[2] F. Gao, C. Chi, Unconditionally stable difference scheme for a onespace dimensional##linear hyperbolic equation, Applied Mathematics and Computation 187 (2007) 12721276. ##[3] A. Saadatmandi, M. Dehghan, Numerical solution of hyperbolic telegraph equation##using the Chebyshev Tau method, Numer. Methods Partial Differential Equations 26##(1) (2010) 239252. ##[4] S.A. Yousefi, Legendre multi wavelet Galerkin method for solving the hyperbolic##telegraph equation, Numerical Method for Partial Differential Equations, (2008).##doi:10.1002/num. ##[5] M. Dehghan, A. Ghesmati, Solution of the secondorder onedimensional hyperbolic##telegraph equation by using the dual reciprocity boundary integral equation (DRBIE)##method, Engineering Analysis with Boundary Elements 34 (2010) 5159. ##[6] S. Das, P.K. Gupta, Homotopy analysis method for solving fractional hyperbolic par##tial differential equations, International Journal of Computer Mathematics 88 (2011)##[7] M.A. Abdou, Adomian decomposition method for solving the telegraph equation in##charged particle transport, J. Quant. Spectrosc. Radiat. Transfer 95 (2005) 407414. ##[8] M. Lakestani, B. N. Saray, Numerical solution of telegraph equation using interpolating##scaling functions, Computers Mathematics with Applications, 60(2010) 19641972. ##[9] R.K. Mohanty, An unconditionally stable difference scheme for the onespace dimen##sional linear hyperbolic equation, Appl. Math. Lett. 17 (2004) 101105. ##[10] R.K. Mohanty, An unconditionally stable finite difference formula for a linear second##order one space dimensional hyperbolic equation with variable coefficients, Appl. Math.##Comput. 165 (2005) 229236. ##[11] L. Lapidus, G.F. Pinder, Numerical Solution of Partial Differential Equations in Science##and Engineering, Wiley, New York, 1982. ##[12] I. Hashim, O. Abdulaziz, S. Momani, Homotopy analysis method for fractional IVPs,##Communications in Nonlinear Science and Numerical Simulation 14 (2009) 674684. ##[13] A. Borhanifar, Reza Abazari, An unconditionally stable parallel difference scheme for##telegraph equation scheme for telegraph equation, Math. Probl. Eng. (2009) Article ID##[14] M. Dehghan, A. Shokri, A numerical method for solving the hyperbolic telegraph equa##tion, Numer. Methods Partial Differential Equations 24 (2008) 10801093. ##[15] M. Dehghan, M. Lakestani, The use of Chebyshev cardinal functions for solution of the ##secondorder onedimensional telegraph equation, Numer. Methods Partial Differential##Equations 25 (2009) 931938. ##[16] J. Biazar, M. Eslami, Analytic solution for Telegraph equation by differential transform##method, Physics Letters A, 374(29)(2010) 29042906. ##[17] L.N. Trefethen, Spectral methods in MATLAB, SIAM, Philadelphia(2000). ##[18] W.S. Don and A. Solomonoff, Accuracy and speed in computing the Chebyshev collo##cation derivative, SIAM J. of Sci. Coput., 16 No. 4(1995) 12531268. ##[19] C. Canuto ,A. Quarteroni, M.Y. Hussaini and T. Zang, Spectral method in fluied me##chanics, SpringerVerlag, New York (1988). ##[20] J.P. Boyd, Chebyshev and Fourier spectral methods, Lecture notes in engineering, 49,##Springerverlag, Berlin(1989). ##[21] R. Baltensperger and M.R. Trummer, Spectral differencing with a twist, SIAM J. of##Sci. Comp., 24,no. 5(2003),14651487. ##[22] R. Baltensperger and J.P. Berrut, The errors in calculating the pseudospectral differen##tiation matrices for ChebyshevGaussLobatto point, Comput. Math. Appl., 37(1999),4148.##]
2stage explicit total variation diminishing preserving RungeKutta methods
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2
In this paper, we investigate the total variation diminishing property for a class of 2stage explicit RungKutta methods of order two (RK2) when applied to the numerical solution of special nonlinear initial value problems (IVPs) for (ODEs). Schemes preserving the essential physical property of diminishing total variation are of great importance in practice. Such schemes are free of spurious oscillations around discontinuities.
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30
38
M.
Mehdizadeh Khalsaraei
M.
Mehdizadeh Khalsaraei
University of Maragheh
University of Maragheh
I. R. Iran
muhammad.mehdizadeh@gmail.com
F.
Khodadosti
F.
Khodadosti
University of Maragheh
University of Maragheh
I. R. Iran
fayyaz64dr@gmail.com
Initial value problem
Method of line
Totalvariationdiminishing
RungKutta methods
[[1] R. Anguelov, Total variation diminishing nonstandard finite difference schemes for conservation laws, J. Math. Comput 51 (2010), 160166. ##[2] M. Mehdizadeh Khalsaraei, An improvement on the positivity results for 2stage explicit RungeKutta methods, J. Comput. Appl. Math 235 (2010), 137143. ##[3] B. Koren, A robust upwind discretization for advection, diffusion and source terms. In: Numerical Methods for AdvectionDiffusion Problems, Notes on Numerical Fluid Mechanics 45 (1993), 117138. ##[4] C.W. Shu, Totalvariationdiminishing time discretizations, SIAM J. Sci. Statist. Comput 9 (1988), 10731084. ##[5] A. Harten, High resolution schemes for hyperbolic conservation laws, SJournal of Computational Physics 49 (1983), 357393. ##[6] W. Hundsdorfer, J. G. Verwer Numerical Solution of TimeDependent Advection DiffusionReaction Equation, Springer (2003)##]
Existence and multiplicity of positive solutions for a coupled system of perturbed nonlinear fractional differential equations
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2
In this paper, we consider a coupled system of nonlinear fractional differential equations (FDEs), such that both equations have a particular perturbed terms. Using emph{LeraySchauder} fixed point theorem, we investigate the existence and multiplicity of positive solutions for this system.
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39
54
Kazem
Ghanbari
Kazem
Ghanbari
Sahand University of
Technology
Sahand University of
Technology
I. R. Iran
Yousef
Gholami
Yousef
Gholami
Sahand University of
Technology
Sahand University of
Technology
I. R. Iran
[[1] B. Ahmad, Juan J. Nieto, Existence results for a coupled system of nonlinear fractional##differential equations with three point boundary conditions, Computers and Mathemat##ics with Applications 58 (2009), 18381843. ##[2] Z. Bai, Existence of solutions for some thirdorder boundaryvalue problems, Electronic##Journal of Differential Equations No. 25 (2008) 16.##54 KAZEM GHANBARI AND YOUSEF GHOLAMI ##[3] K. Ghanbari, Y. Gholami, Existence and multiplicity of positive solutions for mpoint##nonlinear fractional differential equations on the half line, Electronic Journal of Differ##ential equations No. 238 (2012) 115. ##[4] K. Ghanbari, Y. Gholami, Existence and nonexistence results of positive solutions for##nonlinear fractional eigenvalue problem, Journal of Fractional Calculus and Applications##Vol.4 No.2 (2013) 112. ##[5] K. Ghanbari, Y. Gholami, H. Mirzaei, Existence and multiplicity results of positive##solutions for boundary value problems of nonlinear fractional differential equations,##Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical##Analysis 20 (2013) 543558. ##[6] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of fractional##Differential Equations, NorthHolland mathematics studies, Elsvier science 204 (2006). ##[7] K. Q. Lan, Multiple positive solutions of semilinear differential equations with singu##larities, J. Lond. Math. Soc. 63 (2001) 690704. ##[8] K. S. Miller, B. Ross, An Introduction to fractional calculus and fractioal differential##equation, John Wiley, New York (1993). ##[9] I. Poudlobny, Fractional Differential Equations, Mathematics in Science and Applica##tions Academic Press 19 (1999). ##[10] M. ur Rehman, R. A. Khan, A note on boundary value problems for a coupled system##of fractional differential equations, Computers and Mathematics with Applications 61(2011) 26302637. ##[11] S. Zhang, Positive solutions for boundary value problem of nonlinear fractional differ##ential equations, Electronic Journal of Differential Equations 36 (2006) 112. ##[12] S. Liang, J. Zhang, Existence of multiple positive solutions for mpoint fractional bound##ary value problems on an infinite interval, Mathematical and Computer modelling 54##(2011) 13341346. ##[13] S. Zhang, Unbounded solutions to a boundary value problem of fractional order on the##halfline, Computers and mathematics with Applications 61 (2011) 10791087. ##[14] X. Zhao, W. Ge, Unbounde solutions for a fractional boundary value problem on the##infinite interval, Acta Appl Math 109 (2010) 495505. ##[15] S. Zhang, Existence of positive solutions for some class of fractional differential equa##tions, J. Math. Anal. Appl 278 (2003) 136148. ##[16] S. Zhang, G. Han, The existence of a positive solution for a nonlinear fractional differ##ential equation, J. Math. Anal. Appl 252 (2000) 804812. ##[17] X. Zhang, L. Liu, Y. Wu, Multiple positive solutions of a singular fractional differential##equation with negatively perturbed term, Mathematical and Computer Modelling 55##(2012) 12631274. ##[18] Y. Zhang, Z. Bai, T. Feng, Existence results for a coupled system of nonlinear fractional##threepoint boundary value problems at resonance, Computers and Mathematics with##Applications 61 (2011) 10321047.##]
Parameter determination in a parabolic inverse problem in general dimensions
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2
It is well known that the parabolic partial differential equations in two or more space dimensions with overspecified boundary data, feature in the mathematical modeling of many phenomena. In this article, an inverse problem of determining an unknown timedependent source term of a parabolic equation in general dimensions is considered. Employing some transformations, we change the inverse problem to a Volterra integral equation of convolutiontype. By using an explicit procedure based on Sinc function properties, the resulting integral equation is replaced by a system of linear algebraic equations. The convergence analysis is included, and it is shown that the error in the approximate solution is bounded in the infinity norm by the condition number and the norm of the inverse of the coefficient matrix multiplied by a factor that decays exponentially with the size of the system. Some numerical examples are given to demonstrate the computational efficiency of the method.
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55
70
Reza
Zolfaghari
Reza
Zolfaghari
Salman Farsi University of Kazerun
Salman Farsi University of Kazerun
I. R. Iran
rzolfaghari@iust.ac.ir
[[1] P. Amore, A variational Sinc collocation method for StrongCoupling problems, Journal##of Physics A, 39 (22) (2006) 349355.##[2] J.R. Cannon, Y. Lin, Determination of parameter p(t) in Holder classes for some semi##linear parabolic equations, Inverse Problems 4, (1988) 595606. ##[3] J.R. Cannon, Y. Lin, An inverse problem of nding a parameter in a semilinear heat##equation, Journal of Mathematical Analysis and Applications, 145(2) (1990) 470484.##[4] J.R. Cannon, Y. Lin, S. Wang, Determination of source parameter in parabolic equa##tions, Meccanica 27, (1992) 8594.##[5] J.R. Cannon, Y. Lin, Determination of a parameter p(t) in some quasilinear parabolic##dierential equations, Inverse Problems 4, (1988) 3545.##[6] J.R. Cannon, The one dimensional heat equation, 1984 (Reading, MA: AddisonWesley).##[7] M. Dehghan, M. Tatari, Determination of a control parameter in a onedimensional par##abolic equation using the method of radial basis functions, Mathematical and Computer##Modelling 44, (2006) 11601168.##[8] M. Dehghan, An inverse problem of nding a source parameter in a semilinear parabolic##equation, Applied Mathematical Modelling 25, (2001) 743754.##[9] M. Dehghan, A. Saadatmandi, A tau method for the onedimensional parabolic inverse##problem subject to temperature overspecication, Computational Mathematics with Ap##plications, 52 (2006) 933940.##[10] M. Dehghan, Finding a control parameter in onedimensional parabolic equations, Ap##plied Mathematics and Computation, 135 (2003) 491503.##[11] M. Dehghan, Numerical solution of onedimensional parabolic inverse problem, Applied##Mathematics and Computation, 136 (2003) 333344.##[12] M. Dehghan, Determination of a control function in threedimensional parabolic equa##tions, Mathematics and Computers in Simulation, 61 (2003) 89100.##[13] M. Dehghan, Determination of a control parameter in the twodimensional diusion##equation,Applied Numerical Mathematics, 37 (2001) 489502.##[14] M. Dehghan, Fourth order techniques for identing a control parameter in the parabolic##equations, International Journal of Engineering Science, 40 (2002) 433447.##[15] M. Dehghan, Method of lines solutions of the parabolic inverse problem with an over##specication at apoint, Numerical Algorithms, 50 (2009) 417437.##[16] M. Dehghan, Finite dierence schemes for twodimensional parabolic inverse problem##with temperature overspecication, International Journal of Computer Mathematics,##75 (3) (2000) 339349.##[17] F. Li, Z. Wu, Ch. Ye, A nite dierence solution to a twodimensional parabolic inverse##problem,Applied Mathematical Modelling, 36 (2012) 23032313.##[18] Y. Lin, An inverse problem for a cleass of quasilinear parabolic equations, SIAM Journal##on Mathematical Analysis, 22(1) (1991) 146156.##[19] J. Lund, K. Bowers, Sinc methods for quadrature and dierential equations,SIAM,##Philadelphia, 1992.##[20] J. Lund, C. Vogel, A FullyGalerkin method for the solution of an inverse problem in a##parabolic partial dierential equation, Inverse Problems, 6 (1990) 205217.##[21] A.I. Prilepko, D.G. Orlovskii, Determination of the evolution parameter of an equation##and inverse problems of mathematical physics, Part I. Journal of Dierential Equations,##21 (1985) 119129 [and part II, 21 (1985) 694701].##[22] A.I. Prilepko, V.V. Soloev, Solvability of the inverse boundary value problem of nd##ing a coecient of a lower order term in a parabolic equation. Journal of Dierential##Equations, 23(1) (1987) 136143.##[23] W. Rundell, Determination of an unknownnonhomogeneous term in a linear partial##dierential equation from overspecied boundary data, Applicable Analysis, 10 (1980)##[24] A. Shidfar, R. Zolfaghari, J. Damirchi, Application of Sinccollocation method for solv##ing an inverse problem, Journal of Computational and Applied Mathematics, 233 (2009)##[25] A. Shidfar, R. Zolfaghari, Determination of an unknown function in a parabolic in##verse problem by Sinccollocation method,Numerical Methods for Partial Dierential##Equations, 27 (6) (2011) 15841598.##[26] R. Smith, K. Bowers, A SincGalerkin estimation of diusivity in parabolic problems,##Inverse Problems, 9 (1993).##[27] F. Stenger, Numerical methods based on Sinc and analytic functions, Springer, New##York, 1993.##[28] S. Wang, Y. Lin, A nite dierence solution to an invese problem determining a control##function in a parabolic partial dierential equation, Inverse Problems, 5 (1989) 631640.##[29] S. A. Youse, M. Dehghan, Legendre multiscaling functions for solving the one##dimensional parabolic inverse problem, Numerical Methods for Partial Dierential##Equations, 25 (2009) 15021510.##]
The modified simplest equation method and its application
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2
In this paper, the modified simplest equation method is successfully implemented to find travelling wave solutions of the generalized forms $B(n,1)$ and $B(n,1)$ of Burgers equation. This method is direct, effective and easy to calculate, and it is a powerful mathematical tool for obtaining exact travelling wave solutions of the generalized forms $B(n,1)$ and $B(n,1)$ of Burgers equation and can be used to solve other nonlinear partial differential equations in mathematical physics.
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71
77
M.
Akbari
M.
Akbari
University of Guilan
University of Guilan
I. R. Iran
The modified simplest equation method
traveling wave solutions
homogeneous balance
Solitary wave solutions
The generalized forms $B(n
1)$ and $B(n
1)$ of Burgers equation
[[1] E. Fan, Extended tanhfunction method and its applications to nonlinear equations,##Physics Letters A, 277(45) (2000) 212218. ##[2] E. G. Fan, Extended tanhfunction method and its applications to nonlinear equations,##Phys. Lett. A, 277 (2000) 212218. ##[3] E. Fan and H. Zhang, A note on the homogeneous balance method, Phys. Lett. A, 246##(1998) 403406. ##[4] J. H. He and X.H. Wu, Expfunction method and for nonlinear wave equations. Chaos,##Solitons and Fractals, 30 (2006) 700708. ##[5] A. J. M. Jawad, M. D. Petkovic and A. Biswas, Modied simple equation method for##nonlinear evolution equations, Appl. Math. Comput., 217 (2010) 869877. ##[6] S. K. Liu, Z. T. Fu, S. D. Liu and Q. Zhao, Jacobi elliptic function expansion method##and periodic wave solutions of nonlinear wave equatins, Phys. Lett. A, 289 (2001) 7276. ##[7] N. K. Vitanov, Z. I. Dimitrova and H. Kantz, Modied method simplest equation and##application to nonlinear PDFs, Appl. Math. Comput., 216 (2010) 25872595. ##[8] N. K. Vitanov, Modied method simplest equation poerful tool for obtaining exact##and approximate travelingwave solutions of nonlinear PDFs, Commun Nonlinear. Sci.##Numer. Simulat., 16 (2011) 11791185. ##[9] N. K. Vitanov and Z. I. Dimitrova, Application of the method of simplest equation for##obtaining exact travelingwave solutions for two classes of model PDFs from ecoloy and##population dynamics. Commun. Nonlinear Sci. Numer. Simulat., 15 (2010) 28362845. ##[10] M.Wang, X. Li and J. Zhang, The (G′##G )expansion method and travelling wave solutions##of nonlinear evolution equations in mathematical physics, Phys. Lett. A, 372 (2008) 417##[11] M. L. Wang and X. Z. Li, Applications of Fexpansion to periodic wave solutions for a##new Hamiltonian amplitude equation, Chaos Soliton Fract., 24 (2005) 12571268. ##[12] E. M. E. Zayed, A note on the modied simple equation method applied to Sharma##TassoOlver equation, Appl. Math. Comput., 218 (2011) 39623964. ##[13] E. M. E. Zayed and S.A.H. Ibrahim, Exact solutions of nonlinear evolution equations##in mathematical physics using the modied simple equation method, chinese physics##Letters, 29(6) (2012), Article ID 060201. ##[14] H. Zhang, New application of the (G′,G )expansion method, Commun. Nonlinear Sci.##Numer. Simul., 14 (2009) 32203225.##]