Numerical solutions of integral and integro-differential equations using Legendre polynomials

In this paper, a finite Legendre expansion is developed to solve singularly perturbed integral equations, first order integro-differential equations of Volterra type arising in fluid dynamics and Volterra delay integro-differential equations. The error analysis is derived. Numerical results and comparisons with other methods in literature are considered.

[1]  Robert Piessens,et al.  Numerical solution of integral equations of mathematical physics, using Chebyshev polynomials , 1976 .

[2]  W. E. Olmstead,et al.  Singularly perturbed Volterra integral equations II , 1987 .

[3]  Jim M Cushing,et al.  Integrodifferential Equations and Delay Models in Population Dynamics. , 1978 .

[4]  Ernst Hairer,et al.  Numerical methods for singular nonlinear integro-differential equations , 1987 .

[5]  Ray W. Grout,et al.  Interaction of turbulence and scalar fields in premixed flames , 2006 .

[6]  J. B. Scarborough.  Numerical Mathematical Analysis , 1931 .

[7]  Karl Kunisch,et al.  Approximation of the State of Infinite Delay and Volterra-Type Equations , 1983 .

[8]  Takashi Kaminogo.  Continuous dependence of solutions for integrodifferential equations with infinite delay , 1988 .

[9]  C. W. Parkin,et al.  PLASMA FLUX TRANSPORT. , 1969 .

[10]  H. Banks,et al.  Hereditary Control Problems: Numerical Methods Based on Averaging Approximations , 1978 .

[11]  C. Lubich,et al.  Runge-Kutta theory for Volterra and Abel integral equations of the second kind , 1983 .

[12]  Joseph B. Keller,et al.  Temperature of a nonlinearly radiating semi-infinite solid , 1972 .

[13]  Richard Bellman,et al.  Differential-Difference Equations , 1967 .

[14]  Bartur Jumarhon.  High‐order product integration methods for a non‐linear Volterra integro‐differential equation , 1999 .

[15]  S. McKee,et al.  Product Integration Methods for the Nonlinear Basset Equation , 1983 .

[16]  W. E. Olmstead,et al.  Singular perturbation analysis of an integrodifferential equation modelling filament stretching , 1985 .

[17]  J.-P. Kauthen,et al.  A survey of singularly perturbed Volterra equations , 1997 .

[18]  A. B. Basset,et al.  Treatise on Hydrodynamics , 1889, Nature.

[19]  Alvin J. Kay.  Nonlinear integral equations and product integrals , 1975 .

[20]  W. E. Olmstead,et al.  Singular perturbation analysis of a certain volterra integral equation , 1972 .

[21]  A. B. Basset.  A Treatise on Hydrodynamics, Vol. 1 , 1888 .

[22]  W. E. Olmstead,et al.  Asymptotic Solution to a Class of Nonlinear Volterra Integral Equations , 1972 .

[23]  L. Collatz.  The numerical treatment of differential equations , 1961 .

[24]  Peter Linz,et al.  Analytical and numerical methods for Volterra equations , 1985, SIAM studies in applied and numerical mathematics.

[25]  Ernst Hairer,et al.  Fast numerical solution of weakly singular Volterra integral equations , 1988 .

[26]  Olavi Nevanlinna,et al.  Numerical Solution of a Singularly Perturbed Nonlinear Volterra Equation. , 1978 .

[27]  S. McKee,et al.  Weakly Singular Discrete Gronwall Inequalities , 1986 .

[28]  Steven A. Orszag,et al.  Numerical Methods for the Simulation of Turbulence , 1969 .

[29]  G. S. Jordan.  A nonlinear singularly perturbed Volterra integrodifferential equation of nonconvolution type , 1978, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[30]  Hermann Brunner,et al.  Piecewise Polynomial Collocation Methods for Linear Volterra Integro-Differential Equations with Weakly Singular Kernels , 2001, SIAM J. Numer. Anal..

[31]  Nilanjan Chakraborty,et al.  Numerical Investigation of edge flame propagation characteristics in turbulent mixing layers , 2006 .

[32]  Hermann Brunner,et al.  The numerical treatment of Volterra integro-differential equations with unbounded delay , 1989 .

[33]  Ioan Danciu.  Polynomial spline collocation methods for Volterra integro-differential equations , 1996 .

[34]  H. Brunner,et al.  The numerical solution of Volterra equations , 1988 .

[35]  A. S. Lodge.  Body tensor fields in continuum mechanics , 1974 .

[36]  Tao Tang,et al.  A note on collocation methods for Volterra integro-differential equations with weakly singular kernels , 1993 .

[37]  W. E. Olmstead,et al.  Diffusion in a Semi-Infinite Region with Nonlinear Surface Dissipation , 1976 .

[38]  C. Lubich,et al.  Fractional linear multistep methods for Abel-Volterra integral equations of the second kind , 1985 .

[39]  R. K. Miller.  On Volterra’s Population Equation , 1966 .

[40]  J. Fourier.  Théorie analytique de la chaleur , 2009 .

[41]  Hermann Brunner,et al.  Polynomial spline collocation methods for the nonlinear basset equation , 1989 .

[42]  Vito Volterra.  The general equations of biological strife in the case of historical actions , 1939 .

[43]  H. Banks,et al.  Parameter Estimation and Identification for Systems with Delays , 1981 .

[44]  T. A. Burton,et al.  Stability and Periodic Solutions of Ordinary and Functional Differential Equations , 1986 .

[45]  R. Cameron,et al.  Nonlinear Integral Equations , 1964 .

[46]  S. Mckee.  The Analysis of a Variable Step, Variable Coefficient Linear Multistep Method for Solving a Singular Integro–Differential Equation Arising from the Diffusion of Discrete Particles in a Turbulent Fluid , 1979 .

[47]  Hermann Brunner,et al.  The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes , 1985 .

[48]  Robert Piessens,et al.  Modified Clenshaw-Curtis Integration and Applications to Numerical Computation of Integral Transforms , 1987 .

[49]  P. Markowich,et al.  A NONLINEARVOLTERRA INTEGRODIFFERENTIALEQUATION DESCRIBINGTHE STRETCHINGOF POLYMERIC LIQUIDS , 1983 .

[50]  Hermann Brunner,et al.  Collocation methods for nonlinear Volterra integro-differential equations with infinite delay , 1989 .

[51]  S. McKee.  Cyclic Multistep Methods for Solving Volterra Integro-Differential Equations , 1979 .

[52]  V. Lakshmikantham,et al.  Equations with unbounded delay: a survey , 1980 .

[53]  J. B. McLeod,et al.  A nonlinear singularly perturbed Volterra integrodifferential equation occurring in polymer rheology , 1978, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[54]  Michael Renardy,et al.  A Nonlinear Volterra Integrodifferential Equation Describing the Stretching of Polymeric Liquids. , 1983 .

[55]  J. M. Burgers,et al.  Mathematical Examples Illustrating Relations Occurring in the Theory of Turbulent Fluid Motion , 1995 .