Well posedness and numerical solution for a non-local pseudohyperbolic initial boundary value problem

References

• A. Ashyralyev and N. Aggez, Finite difference method for hyperbolic equations with the nonlocal integral condition, Discrete Dyn. Nat. Soc. 2011 (2011), pp. 1–15.
   Web of Science ®Google Scholar
• J.R. Cannon, The solution of heat equation subject to the specification of energy, Quart. Appl. Math. 21(2) (1963), pp. 155–160.
   Google Scholar
• J.R. Cannon, S.P. Esteva, and J. van der Hoek, A Galerkin procedure for the diffusion equation subject to the specification of mass, SIAM J. Numer. Anal. 24(3) (1987), pp. 449–515.
   Web of Science ®Google Scholar
• J.R. Cannon and J.R. Lin, An inverse problem of finding a parameter in a semi-linear heat equation, J. Math. Anal. Appl. 145 (1990), pp. 470–484.
   Web of Science ®Google Scholar
• Y.S. Choi and K.-Y. Chan, A parabolic equation with nonlocal boundary conditions arising from electrochemistry, Nonlinear Anal. 18 (4) (1992), pp. 317–331.
   Web of Science ®Google Scholar
• M. Dehghan, On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation, Numer. Methods Partial Differential Equations 21(1) (2004), pp. 24–40.
   Web of Science ®Google Scholar
• M. Dehghan, The solution of a nonclassic problem for one-dimensional hyperbolic equation using the decomposition procedure, Int. J. Comput. Math. 81(8) (2004), pp. 979–989.
   Web of Science ®Google Scholar
• M. Dehghan and M. Lakestani, The use of cubic B-spline scaling functions for solving the one-dimensional hyperbolic equation with a nonlocal conservation condition, Numer. Methods Partial Differential Equations 23(6) (2007), pp. 1277–1289.
   Web of Science ®Google Scholar
• R.E. Ewing and T. Lin, A class of parameter estimation techniques for fluid flow in porous media, Adv. Water Res. 14(2) (1991), pp. 89–97.
   Web of Science ®Google Scholar
• L. Garding, Cauchy Problem for Hyperbolic Equations, Lecture Notes, University of Chicago, 1957.
   Google Scholar
• W. Jiang and M. Cui, Solving nonlinear singular pseudoparabolic equations with nonlocal mixed conditions in the reproducing kernel space, Int. J. Comput. Math. 87(15) (2010), pp. 3430–3442.
   Web of Science ®Google Scholar
• O.M. Korpusov, Blow-up in Nonclassical Wave Equations, URSS, Moscow, 2010.
   Google Scholar
• M.X. Li, H1-Galerkin expanded mixed finite element methods for pseudo-hyperbolic equations, Adv. Mater. Res. 268–270 (2011), pp. 908–912.
   Google Scholar
• S. Mesloub and M.R. Aboelrish, On an evolution-mixed thermoelastic system problem, Int. J. Comput. Math. 92(2) (2015), pp. 424–439.
   Web of Science ®Google Scholar
• S. Mesloub and A. Bouziani, Mixed problem with integral conditions for a certain class of hyperbolic equations, J. Appl. Math. 1(3) (2001), pp. 107–116.
   Google Scholar
• S. Mesloub and S. Obaidat, On the application of the homotopy analysis method for a nonlocal mixed problem with Bessel operator, Appl. Math. Comput. 219(8) (2012), pp. 3477–3485.
   Web of Science ®Google Scholar
• G. Ponce, Global existence of small of solutions to a class of nonlinear evolution equations, Nonlinear Anal. 9 (1985), pp. 399–418.
   Web of Science ®Google Scholar
• G. Ponce and R. Racke, Global existence of small of solutions to the initial value problem for nonlinear thermoelasticity, J. Differ. Equ. 87 (1990), pp. 70–83.
   Web of Science ®Google Scholar
• S.V. Potapova, Boundary value problems for pseudohyperbolic equations with variable time direction, J. Pure Appl.Math. 3(1) (2012), pp. 75–91.
   Google Scholar
• L.S. Pulkina, Solution to nonlocal problems of pseudohyperbolic equations, Electron. J. Differ. Equ. 2012(116) (2012), pp. 1–9.
   Google Scholar
• A. Saadatmandi and M. Dehghan, Numerical solution of the one-dimensional wave equation with an integral condition, Numer. Methods Partial. Differential Equations 23(2) (2007), pp. 282–292.
   Web of Science ®Google Scholar
• A.A. Samarskii, On certain problems of the modern theory of differential equations, Differ. Uravn. 16(11) (1980), pp. 1221–1228.
   Google Scholar
• P. Shi, Weak solution to an evolution problem with a nonlocal constraint, SIAM. J. Math. Anal. 24(1) (1993), pp. 46–58.
   Web of Science ®Google Scholar
• P. Shi and M. Shillor, On design of contact patterns in one-dimensional thermoelasticity, Theoretical Aspects of Industrial Design (Wright-Patterson Air Force Base, OH, 1990), SIAM, Pennsylvania, 1992, pp. 76–82.
   Google Scholar
• E. Shivanian and H.R. Khodabandehlo, Meshless local radial point interpolation (MLRPI) on the telegraph equation with purely integral conditions, Eur. Phys. J. Plus 129(11) (2014), pp. 241.
   Web of Science ®Google Scholar
• G.D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd ed, Clarendon Press, Oxford, 1985.
   Google Scholar