Abstract
This work deals with approximation solutions to a type of integro-differential equations in several complex variables. It concerns the Cauchy formula on higher dimensional domains. In our study, we make use of multiple power series expansions and an iterative computation method to solve a kind of integro-differential equation. We introduce a symmetrized topology product area which is called a bicylinder. We expand functions and derivatives of them to power series. Moreover we obtain unknown functions by comparing coefficients of the series on both sides of equations. We express the approximation solutions by a regular product of matrixes.
Similar content being viewed by others
References
Chen L P, Zhong T D, Qian T. High order boundary integral formula and integrodifferential equation on Stein manifolds. Complex Analysis and Operator Theory, 2012, 6(2): 447–464
Chen L P, Zhong T D. Higher order singular integral equations on complex hypersphere. Acta Mathematica Scientia, 2010, 30B(5): 1785–1792
Chen L P, Zhong T D. Regularization for higher order singular integral equations. Integral Equations and Operator Theory, 2008, 62(1): 65–76
Zhong T D, Chen L P. The Poincaré-Bertrand formula for the Bochner-Martinelli integral. Integral Equations and Operator Theory, 2006, 54(4): 585–595
Zhong T D. Integral Representations for Several Complex Variables and Multidimensional Singular Integral Equations (in Chinese). Xiamen: Xiamen University Press, 1986
Qian T, Zhong T D. The differential integral equations on smooth closed orientable manifolds. Acta Mathematica Scientia, 2001, 21B(1): 1–8
Leiterer J. Lecture on Cauchy's Problem in Linear Partial Differential Equations. New York: Dover Publications, 1952
Henkin G M, Leiterer J. Theory of Functions on Complex Manifolds. Berlin and Boston: Akademie-Verlag and Birkhäuser-Verlag, 1984
Yüzbasi S Y, Sahin N Y, Yildirm A M. A collocation approach for solving high-order linear fredholmvolterra integro-differential equations. Mathematical and Computer Modelling, 2012, 55(3/4): 547–563
Xiang S H, He K X. On the implementation of discontinuous Galerkin method for volterra integral equations with highly oscillatory Bessel kernels. Applied Mathematics and Computation, 2013, 219(9): 4884–4891
Eskhuvatov Z K, Ahmedov A, Nik Long N M A, Amalina N J. Approximating Cauchy-type singular integral by an automatic quadrature scheme. Journal of Computational and Applied Mathematics, 2011, 235(16): 4675–4686
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the National Natural Science Foundation of China (11771357, 11171277), the Fundamental Research Funds for the Central Universities of Xiamen University (2010121002), the Science Foundation of Fujian province of China (S0850029, 2008J0206).
Rights and permissions
About this article
Cite this article
Chen, L. The Approximation Solutions for Higher Dimensional Integro-Differential Equations. Acta Math Sci 39, 1309–1318 (2019). https://doi.org/10.1007/s10473-019-0509-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10473-019-0509-7