A super accurate shifted Tau method for numerical computation of the Sobolev-type differential equation with nonlocal boundary conditions

In this article, we propose a super accurate numerical scheme to solve the one-dimensional Sobolev type partial differential equation with an initial and two nonlocal integral boundary conditions. Our proposed methods are based on the shifted Standard and shifted Chebyshev Tau method. Firstly, We convert the model of partial differential equation to a linear algebraic equation and then we solve this system. Shifted Standard and shifted Chebyshev polynomials are applied for giving the computational results. Numerical results are presented for some problems to demonstrate the usefulness and accuracy of this approach. The method is easy to apply and produces very accurate numerical results.

[1]  S. Shahmorad,et al.  Numerical solution of the general form linear Fredholm-Volterra integro-differential equations by the Tau method with an error estimation , 2005, Appl. Math. Comput..

[2]  Said Mesloub,et al.  On a class of singular hyperbolic equation with a weighted integral condition , 1999 .

[3]  A. Bouziani,et al.  SOLUTION TO A SEMILINEAR PSEUDOPARABOLIC PROBLEM WITH INTEGRAL CONDITIONS , 2006 .

[4]  Hosseini Seyed Mohammad,et al.  Tau numerical solution of Fredholm integro-differential equations with arbitrary polynomial bases , 2003 .

[5]  Heinz Brill,et al.  A semilinear Sobolev evolution equation in a Banach space , 1977 .

[6]  K. Liu,et al.  Tau method approximation of differential eigenvalue problems where the spectral parameter enters nonlinearly , 1987 .

[7]  H. G. Khajah,et al.  Iterated solutions of linear operator equations with the Tau method , 1997, Math. Comput..

[8]  E. Ortiz,et al.  A bidimensional tau-elements method for the numerical solution of nonlinear partial differential equations with an application to burgers' equation , 1986 .

[9]  Ralph E. Showalter,et al.  Existence and Representation Theorems for a Semilinear Sobolev Equation in Banach Space , 1972 .

[10]  Analytical Solutions of the Slip Magnetohydrodynamic Viscous Flow over a Stretching Sheet by Using the Laplace–Adomian Decomposition Method , 2012 .

[11]  George W. Batten,et al.  Second-Order Correct Boundary Conditions for the Numerical Solution of the Mixed Boundary Problem for Parabolic Equations , 1963 .

[12]  Mohsen Razzaghi,et al.  Rational Chebyshev tau method for solving Volterra's population model , 2004, Appl. Math. Comput..

[13]  Eduardo L. Ortiz,et al.  An operational approach to the Tau method for the numerical solution of non-linear differential equations , 1981, Computing.

[14]  J. R. Cannon,et al.  The solution of the heat equation subject to the specification of energy , 1963 .

[15]  B. Soltanalizadeh.  An Approximation Method Based on Matrix Formulated Algorithm for the Numerical Study of a Biharmonic Equation , 2011 .

[16]  K. Balachandran,et al.  SOBOLEV TYPE INTEGRODIFFERENTIAL EQUATION WITH NONLOCAL CONDITION IN BANACH SPACES , 2003 .

[17]  M. Dehghan.  The one-dimensional heat equation subject to a boundary integral specification , 2007 .

[18]  Shruti A. Dubey.  NUMERICAL SOLUTION FOR NONLOCAL SOBOLEV-TYPE DIFFERENTIAL EQUATIONS , 2010 .

[19]  On the Analytic Solution for a Steady Magnetohydrodynamic Equation , 2013 .

[20]  Saeid Abbasbandy,et al.  A matrix formulation to the wave equation with non-local boundary condition , 2011, Int. J. Comput. Math..

[21]  S. Shahmorad,et al.  Numerical solution of the nonlinear Volterra integro-differential equations by the Tau method , 2007, Appl. Math. Comput..

[22]  Babak Soltanalizadeh,et al.  Numerical solution of a nonlinear singular Volterra integral system by the Newton product integration method , 2013, Math. Comput. Model..

[23]  B. Soltanalizadeh.  APPLICATION OF DIFFERENTIAL TRANSFORMATION METHOD FOR SOLVING A FOURTH-ORDER PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS , 2012 .

[24]  Babak Soltanalizadeh,et al.  Numerical solution for system of singular nonlinear Volterra integro-differential equations by Newton-Product method , 2013, Appl. Math. Comput..

[25]  B. Soltanalizadeh.  Differential transformation method for solving one-space-dimensional telegraph equation , 2011 .

[26]  Abdelfatah Bouziani,et al.  Strong solution for a mixed problem with nonlocal condition for certain pluriparabolic equations , 1997 .

[27]  C. Lanczos,et al.  Trigonometric Interpolation of Empirical and Analytical Functions , 1938 .

[28]  Mehdi Dehghan,et al.  The method of lines for solution of the one-dimensional wave equation subject to an integral conservation condition , 2008, Comput. Math. Appl..

[29]  Muhammad Akram,et al.  A NUMERICAL METHOD FOR THE HEAT EQUATION WITH A NONLOCAL BOUNDARY CONDITION , 2005 .

[30]  Graeme Fairweather,et al.  The Reformulation and Numerical Solution of Certain Nonclassical Initial-Boundary Value Problems , 1991, SIAM J. Sci. Comput..

[31]  Mehdi Dehghan,et al.  Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method , 2010 .