Abstract
Using the Picard iteration method and treating the involved integration by numerical quadrature formulas, we propose a numerical scheme for the second kind nonlinear Volterra integral equations. For enlarging the convergence region of the Picard iteration method, multistage algorithm is devised. We also introduce an algorithm for problems with some singularities at the limits of integration including fractional integral equations. Numerical tests verify the validity of the proposed schemes.
Highlights
The Volterra integral equations arise in many scientific and engineering fields such as the population dynamics, spread of epidemics, semi-conductor devices, vehicular traffic, the theory of optimal control, the kinetic theory of gases and economics [1]-[7]
The initial or boundary value problems for ordinary differential equations and some fractional differential equations can be equivalently expressed by the second-kind Volterra integral equation [6]-[9]
The specific conditions under which a solution exists for the nonlinear Volterra integral equation are considered in [1]-[4] [7]
Summary
The Volterra integral equations arise in many scientific and engineering fields such as the population dynamics, spread of epidemics, semi-conductor devices, vehicular traffic, the theory of optimal control, the kinetic theory of gases and economics [1]-[7]. We consider the general nonlinear Volterra integral equation of the second kind u (t ) = C (t ) + ∫tt0 f (t, s,u (s)) ds, t ≥ t0 ,. How to cite this paper: Chen, L. and Duan, J.S. (2015) Multistage Numerical Picard Iteration Methods for Nonlinear Volterra Integral Equations of the Second Kind. The Picard iteration method, or the successive approximations method, is a direct and convenient technique for the resolution of differential equations This method solves any problem by finding successive approximations to the solution by starting with the zeroth approximation. We concern on the numerical Picard iteration methods for nonlinear Volterra integral Equation (1). By using the proposed methods, we treat the involved integrals numerically and enlarge the effective region of convergence of the Picard iteration. To show the effectiveness of the proposed algorithms, we perform some numerical results
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