# Numerical Analysis of the One-Demential Wave Equation Subject to a Boundary Integral Specification

## DOI:

https://doi.org/10.48048/wjst.2018.1153## Keywords:

Wave equation, nonlocal boundary conditions, expansion methods, matrix formulation## Abstract

In this paper a numerical technique is developed for the one-dimensional wave equation that combines classical and integral boundary conditions. A new matrix formulation technique with arbitrary polynomial bases is proposed for the analytical solution of this kind of partial differential equation. Not only have the exact solutions been achieved by the known forms of the series solutions, but also, for the finite terms of series, the corresponding numerical approximations have been computed. We give a simple and efficient algorithm based on an iterative process for numerical solution of the method. Some numerical examples are included to demonstrate the validity and applicability of the technique.

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## References

EL Ortiz and H Samara. An operational approach to the Tau method for the numerical solution of nonlinear differential equations. Computing 1981; 27, 15-25.

EL Ortiz. The Tau method. SIAM J. Numer. Anal. 1969; 6, 480-92.

D Gottlieb, M Hussaini and S Orszag. Theory and Applications of Spectral Methods. In: R Voigt (ed.). Spectral Methods for Partial Differential Equations. SIAM, Philadelphia, 1984.

KM Liu and EL Ortiz. Eigenvalue problems for singularly perturbed differential equations. In: Proceedings of the BAIL II Conference. Boole Press, Dublen, 1982, p. 324-9.

KM Liu and EL Ortiz. Numerical solution of eigenvalue problems for partial differential equations with the Tau-lines Method. Comput. Math. Appl. 1986; 12, 1153-68.

KM Liu, EL Ortiz and KS Pun. Numerical Solution of Steklov’s Partial Differential Equation Eigenvalue Problem. In: JJH Miller (ed.). Computational and Asymptotic Methods for Boundary and Interior Layers (III), Boole Press, Dublin, 1984, p. 244-9.

B Soltanalizadeh, HR Ghehsareh and S Abbasbandy. A super accurate shifted Tau method for numerical computation of the Sobolev-type differential equation with non-local boundry conditions, Appl. Math. Comput. 2014; 236, 683-92.

B Soltanalizadeh, HR Ghehsareh and S Abbasbandy. Development of the Tau method for the numerical study of a fourth-order parabolic partial differential equation. UPB Sci. Bull. Ser. A 2013; 75, 165-76.

S Abbasbandy and E Taati. Numerical solution of system of nonlinear Volterra integro-differential equationswith nonlinear differential part by the operational Tau method and error estimation. J. Comput. Appl. Math. 2009; 231, 106-13.

JR Cannon. The solution of the heat equation subject to the specification of energy. Quart. Appl. Math. 1963; 21, 155-60.

JG Batten. Second-order correct boundary conditions for the numerical solution of the mixed boundary problem for parabolic equations. Math. Comput. 1963; 17, 405-13.

LI Kamynin. A boundary value problem in the theory of the heat conduction with nonclassical boundary condition. Z. Vychisl. Mat. Fiz. 1964; 4, 1006-24.

NI Ionkin. Solutions of boundary value problem in heat conductions theory with nonlocal boundary conditions. Diff. Uravn. 1977; 13, 294-304.

SA Beilin. Existence of solutions for one-dimensional wave equations with nonlocal conditions. Electron. J. Diff. Eq. 2001; 76, 1-8.

A Bouziani. Strong solution for a mixed problem with nonlocal condition for certain pluriparabolic equations. Hiroshima Math. J. 1997; 27, 373-90.

M Dehghan. On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation. Numer. Meth. Part. Diff. Eq. 2005; 21, 24-40.

M Dehghan and A Saadatmandi. Variational iteration method for solving the wave equation subject to an integral conservation condition. Chaos Solitons Fractals 2009; 41, 1448-53

LS Pulkina. On Solvability in L2 of Nonlocal Problem with Integral Conditions for a Hyperbolic Equation. Differentsialnye Uravneniya, 2000.

W Ang. A numerical method for the wave equation subject to a non-local conservation condition. Appl. Numer. Math. 2006; 56, 1054-60.

B Soltanalizadeh. Numerical analysis of the one-dimensional heat equation subject to a boundary integral specification. Opt. Commun. 2011; 284, 2108-12.

HR Ghehsareh, B Soltanalizadeh and S Abbasbandy. A matrix formulation to the wave equation with nonlocal boundary condition. Int. J. Comput. Math. 2011; 88, 1681-96.

B Soltanalizadeh. Differential transformation method for solving one-space-dimensional Telegraph equation. Comput. Appl. Math. 2011; 30, 639653

B Soltanalizadeh and A Yildirim. Application of differential transformation method for numerical computation of Regularized Long wave equation. Z. Naturforsch. 2012; 67, 160 -6.

Z Sarmast, MR Safi and E Farzaneh. Post olptimality analysis in fuzzy multi objective linear programming. Int. J. Math. Appl. 2018; 6 1, 925-37.

R Abazari and M Ganji. Extended two-dimensional DTM and its application on nonlinear PDEs with proportional dela. Int. J. Comput. Math. 2011; 88, 1749-62.

R Abazari. Application of (G/G ́ ) expansion method to travelling wave solutions of three nonlinear evolution equation. Comput. Fluids 2010; 39, 1957-63.

R Hekmati. On sensitivity of iterated methods to initial points in solving nonlinear system of equations and predict methods behavior far away from solution. Aust. J. Basic Appl. Sci. 2013; 7, 317-27.

R Abazari and B Soltanalizadeh. Reduced differential transform method and its application on Kawa- hara equations. Thai J. Math. 2013; 11, 199-216.

B Soltanalizadeh. Application of differential transformation method for solving a fourth-order parabolic partial differential equations. Int. J. Pure Appl. Math. 2012; 78, 299-308.

R Hekmati. On efficiency of non-monotone adaptive trust region and scaled trust region methods in solving nonlinear systems of equations. Control and Optimization in Applied Math. 2016; 1, 31-40.

S Kumar, XB Yin and D Kumar. A modified homotopy analysis method for solution of fractional wave equations. Adv. Mech. Eng. 2015; 7, 1-8.

S Kumar, D Kumar and J Singh. Fractional modelling arising in unidirectional propagation of long waves in dispersive media. Adv. Nonlinear Anal. 2016; 5, 383-94.

Z Sarmast, B Soltanalizadeh and K Boubaker. A new numerical method to study a Second-order hyperbolic equation. South Asian J. Math. 2014; 6, 285-96.

F Shakeri and M Dehghan. The method of lines for solution of the one-dimensional wave equation subject to an integral conservation condition. J. Comp. Math. Appl. 2008; 56, 2175-88.

A Saadatmandi and M Dehghan. Numerical solution of the one-dimensional wave equation with an integral condition. Numer. Meth. Partial Diff. Eq. 2007; 23, 282-92.

M Dehghan. On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation. Numer. Meth. Partial Diff. Eq. 2005; 21, 24-40.

A Yazdani and E Boerwinkle. Causal inference in the age of decision medicine. J. Data Mining Genomic Proteomics 2015; 6, 163.

A Yazdani and E Boerwinkle. Causal inference at the population level. Int. J. Res. Med. Sci. 2014; 2, 1368-70.

A Yazdani, A Yazdani and E Boerwinkle. A causal network analysis of the fatty acid metabolome in African-Americans reveals a critical role for palmitoleate and margarate. J. Integrat. Biol. 2016; 20, 480-4.

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*Walailak Journal of Science and Technology (WJST)*,

*15*(6), 421–437. https://doi.org/10.48048/wjst.2018.1153

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