Exact and approximation product solutions form of heat equation with nonlocal boundary conditions using Ritz–Galerkin method with Bernoulli polynomials basis

In this article, a new method is introduced for finding the exact solution of the product form of parabolic equation with nonlocal boundary conditions. Approximation solution of the present problem is implemented by the Ritz–Galerkin method in Bernoulli polynomials basis. The properties of Bernoulli polynomials are first presented, then Ritz–Galerkin method in Bernoulli polynomials is used to reduce the given differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the techniques presented in this article for finding the exact and approximation solutions. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1143–1158, 2017

[1]  S. Mesloub.  On a singular two dimensional nonlinear evolution equation with nonlocal conditions , 2008 .

[2]  Fulya Callialp Kunter,et al.  Radially symmetric weighted extended b-spline model , 2011, Appl. Math. Comput..

[3]  Whye-Teong Ang,et al.  Numerical solution of a non-classical parabolic problem: An integro-differential approach , 2006, Appl. Math. Comput..

[4]  Abba B. Gumel,et al.  On the numerical solution of the diffusion equation subject to the specification of mass , 1999, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[5]  William Alan Day.  Parabolic equations and thermodynamics , 1992 .

[6]  John R. Cannon,et al.  A class of non-linear non-classical parabolic equations , 1989 .

[7]  M. Razzaghi,et al.  Hybrid functions approach for optimal control of systems described by integro-differential equations , 2013 .

[8]  Mohsen Razzaghi,et al.  Linear quadratic optimal control problems via shifted Legendre state parametrization , 1994 .

[9]  Mehdi Dehghan,et al.  Ritz‐Galerkin method with Bernstein polynomial basis for finding the product solution form of heat equation with non‐classic boundary conditions , 2012 .

[10]  M. Dehghan.  Efficient techniques for the second-order parabolic equation subject to nonlocal specifications , 2005 .

[11]  Fulya Callialp Kunter,et al.  3D web-splines solution to human eye heat distribution using bioheat equation , 2011 .

[12]  Lin Yanping,et al.  An implicit finite difference scheme for the diffusion equation subject to mass specification , 1990 .

[13]  Yadollah Ordokhani,et al.  A numerical solution for fractional optimal control problems via Bernoulli polynomials , 2016 .

[14]  Mehdi Dehghan,et al.  Use of radial basis functions for solving the second‐order parabolic equation with nonlocal boundary conditions , 2008 .

[15]  M. Dehghan.  A computational study of the one‐dimensional parabolic equation subject to nonclassical boundary specifications , 2006 .

[16]  Mohsen Razzaghi,et al.  HYBRID FUNCTIONS APPROACH FOR LINEARLY CONSTRAINED QUADRATIC OPTIMAL CONTROL PROBLEMS , 2003 .

[17]  Avner Friedman,et al.  Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions , 1986 .

[18]  Gunnar Ekolin,et al.  Finite difference methods for a nonlocal boundary value problem for the heat equation , 1991 .

[19]  Felix E. Browder,et al.  The One-Dimensional Heat Equation: Preface , 1984 .

[20]  Mohsen Razzaghi,et al.  A tau method approach for the diffusion equation with nonlocal boundary conditions , 2004, Int. J. Comput. Math..

[21]  Lin Yanping,et al.  A numerical method for the diffusion equation with nonlocal boundary specifications , 1990 .

[22]  A. Bouziani,et al.  Galerkin method applied to a parabolic evolution problem with nonlocal boundary conditions , 2008 .

[23]  Mehdi Dehghan,et al.  On the solution of the non-local parabolic partial differential equations via radial basis functions , 2009 .

[24]  General nonlocal nonlinear boundary value problem for differential equation of 3rd order , 1997 .

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

[26]  Mehdi Dehghan,et al.  The operational matrices of Bernstein polynomials for solving the parabolic equation subject to specification of the mass , 2011, J. Comput. Appl. Math..

[27]  Mehdi Dehghan.  Numerical solution of a parabolic equation with non-local boundary specifications , 2003, Appl. Math. Comput..

[28]  Hong Du,et al.  The solution of a parabolic differential equation with non-local boundary conditions in the reproducing kernel space , 2008, Appl. Math. Comput..

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

[30]  M. Dehghan,et al.  Composite spectral method for solution of the diffusion equation with specification of energy , 2008 .

[31]  Zhi‐zhong Sun.  A second-order accurate finite difference scheme for a class of nonlocal parabolic equations with natural boundary conditions , 1996 .

[32]  Minggen Cui,et al.  Numerical algorithm for parabolic problems with non-classical conditions , 2009 .

[33]  Yadollah Ordokhani,et al.  Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations , 2014 .