Abstract
In this paper, a sequence of approximate solution converging uniformly to the exact solution for a class of integro-differential equation with an integral boundary condition arising in chemical engineering, underground water flow and population dynamics and other field of physics and mathematical chemistry is obtained by using an iterative method. Its exact solution is represented in the form of series in the reproducing kernel space. The n-term approximation u n (x) is proved to converge to the exact solution u(x). Moreover, the first derivative of u n (x) is also convergent to the first derivative of u(x).
Similar content being viewed by others
References
Gallardo J.M.: Second order differential operators with integral boundary conditions and generation of semigroups. Rocky Mt. J. Math. 30, 1265–1292 (2000)
Karakostas G.L., Tsamatos P.Ch.: Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems. Electron. J. Differ. Equ. 30, 1–17 (2002)
Lomtatidze A., Malaguti L.: On a nonlocal boundary-value problems for second order nonlinear singular differential equations. Georgian Math. J. 7, 133–154 (2000)
Bouziani A., Merazga N.: Solution to a semilinear pseudoparabolic problem with integral conditions. Electron. J. Differ. Equ. 115, 1–18 (2006)
Merazga N., Bouziani A.: On a time-discretization method for a semilinear heat equation with purely integral conditions in a nonclassical function space. Nonlinear Anal. 66, 604–623 (2007)
Pulkina L.S.: A non-local problem with integral conditions for hyperbolic equations. Electron. J. Differ. Equ. 45, 1–6 (1999)
Jankowski T.: Monotone and numerical-analytic methods for differential equations. Comput. Math. Appl. 45, 1823–1828 (2003)
Cui M.G., Chen Z.: The exact solution of nonlinear age-structured population model. Nonlinear Anal. Real World Appl. 8, 1096–1112 (2007)
Capobianco E.: Kernel methods and flexible inference for complex stochastic dynamics. Physica A: Stat. Mech. Appl. 387, 4077–4098 (2008)
Ball J.A., Bolotnikov V., Fang Q.: Schur-class multipliers on the Arveson space: De Branges-Rovnyak reproducing kernel spaces and commutative transfer-function realizations. J. Math. Anal. Appl. 341, 519–539 (2008)
Chen Z., Lin Y.Z.: The exact solution of a linear integral equation with weakly singular kernel. J. Math. Anal. Appl. 344, 726–734 (2008)
G.E. Fasshauer, Meshfree approximation methods with MATLAB. With 1 CD-ROM (Windows, Macintosh and UNIX). (World Scientific Publishing Co. Pte. Ltd, Hackensack, NJ, 2007)
Cui M.G., Lin Y.Z.: Nonlinear numercial analysis in the reproducing kernel space. Nova Science Publisher, New York (2008)
Cui M.G., Geng F.Z.: Solving singular two-point boundary value problem in reproducing kernel space. J. Comput. Appl. Math. 205, 6–15 (2007)
Yildirim A.: Solution of BVPs for fourth-order integro-differential equations by using homotopy perturbation method. Comput. Math. Appl. 56, 3175–3180 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported by the NSF (No. 40871082) of China, the NSF (No. A2007-11) of Heilongjiang Province and the Dr. Fund of Harbin Normal University (No. KGB200901).
Rights and permissions
About this article
Cite this article
Yao, H. New algorithm for the numerical solution of the integro-differential equation with an integral boundary condition. J Math Chem 47, 1054–1067 (2010). https://doi.org/10.1007/s10910-009-9628-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-009-9628-z