In scientific and engineering problems Volterra integral equations are always encountered. Applications of Volterra integral equations arise in areas such as population dynamics, spread of epidemics in the society, etc. The problem statement is to obtain a good numerical solution to such an integral equation.
A brief theory of Volterra Integral equation, particularly, of weakly singular types, and a numerical method, the collocation method, for solving such equations, in particular Volterra integral equation of second kind, is handled in this paper. The principle of this method is to approximate the exact solution of the equation in a suitable finite dimensional space. The approximating space considered here is the polynomial spline space. In the treatment of the collocation method emphasis is laid, during discretization, on the mesh type. The approximating space applied here is the polynomial spline space. The discrete convergence properties of spline collocation solutions for certain Volterra integral equations with weakly singular kernels shall is analyzed. The order of convergence of spline collocation on equidistant mesh points is also compared with approximation on graded meshes. In particular, the attainable convergence orders at the collocation points are examined for certain choices of the collocation parameters.
Abstract: Volterra integral equations of weakly singular types have solutions which are non-smooth near the initial point of the integration interval. The implementation of the collocation spline method will lead us to the examination of the attainable order of convergence of this method on graded mesh points for non linear Volterra integral equations with singular kernels.
Keywords: Weakly singular Volterra integral equation; Numerical method: Collocation method
Content
1. Introduction.. 2
2. Volterra integral equations with weakly singular kernel.. 2
2.1 Existence and uniqueness of the solution.. 3
3. Numerical Method – the Collocation method.. 5
3.1 The approximating spline spaces.. 5
3.2 Discretization of the collocation equation.. 6
4. Order of Convergence.. 10
5. Numerical examples.. 12
6. References.. 19
1. Introduction
In scientific and engineering problems Volterra integral equations are always encountered. Applications of Volterra integral equations arise in areas such as population dynamics, spread of epidemics in the society, heat and fluid conductivity, etc. Some classes of initial boundary problems of heat conductivity have been transformed to an equivalent system of Volterra type integral equations of second type (Jalalvand, Jazbi, & Mokhtarzadeh, 2013). The problem statement is to obtain a good numerical solution to such an integral equation.
Numerical solutions of second kind Volterra integral equations with weakly singular kernels have been investigated by Brunner (1985), Brunner, Pedas & Vainikko (1997), Brunner and van der Houwen (1986), Linz (1985), Makroglou (1981), Te Riele (1982), Diogo and Lima (2007).
IIn this paper we shall present a brief theory of Volterra Integral equation, particularly, of weakly singular types. We are interested in finding an approximate solution which exhibits high order of convergence. This will lead us to the implementation of an approximation with polynomial splines on special graded meshes. This implementation will be carried out along the lines of Brunner (1985), Brunner and van der Houwen (1986), Te Riele (1982) and Brunner, Pedas & Vainikko (1997). The principle of this method is to approximate the exact solution of the equation in a suitable finite dimensional space.
2. Volterra integral equations with weakly singular kernel
An integral equation is defined to be a functional equation, whereby the unknown function features under the integral sign. A Volterra integral equation, an equation where the unknown function appears under the integral sign, is characterized by a variable upper limit and a fixed lower limit of integration. Volterra integral equations are classified in three main groups: first, second and third kind. In this paper we shall be dealing with (nonlinear) Volterra integral equation of second kind of the form:
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