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.
Table of Contents
1. Introduction
2. Volterra integral equations with weakly singular kernel
2.1 Existence and uniqueness of the solution
3. Numerical Method – the Collocation method
3.1 The approximating spline spaces
3.2 Discretization of the collocation equation
4. Order of Convergence
5. Numerical examples
6. References
Research Objectives and Themes
The primary objective of this work is to develop and implement an efficient numerical approximation for non-linear Volterra integral equations of the second kind that feature weakly singular kernels. The research investigates how polynomial splines applied on specially graded meshes can achieve a high order of convergence, addressing the non-smooth behavior of solutions near the initial point of the integration interval.
- Numerical analysis of Volterra integral equations with weakly singular kernels.
- Implementation of the collocation spline method for approximating exact solutions.
- Examination of convergence rates using uniform and asymmetrically graded meshes.
- Product integration techniques for the discretization of collocation equations.
- Validation of the numerical method through practical examples and error analysis.
Excerpt from the Book
3. Numerical Method – the Collocation method
In this section we shall handle a numerical method – the collocation method – for solving weakly singular integral equations of the form (2.2). In the collocation method we approximate the exact solution of a given functional equation in a suitable chosen finite subset of the interval on which the equation is to be solved. The approximating spaces applied to this problem are certain polynomial spline spaces.
3.1 The approximating spline spaces
The weakly singular Volterra integral equation (2.2) will be solved in the interval I := [0, T]. For a given N ∈ N let ∏N := {t0,t1,…,tN} with (0= t0 < t1 < … < tN = T) be the mesh of the interval I. ZN := {tk : 1 ≤ k ≤ N – 1} denotes the set of the interior points of the partition ∏N, and associates as well the subinterval σ0 := [t0, t1], σk := (tk, tk+1] for k = 1, …,N-1.
Summary of Chapters
1. Introduction: This chapter provides an overview of the significance of Volterra integral equations in scientific and engineering fields and outlines the research objective of achieving high-order numerical convergence.
2. Volterra integral equations with weakly singular kernel: This section defines the specific class of nonlinear Volterra integral equations under study and establishes the mathematical criteria for the existence and uniqueness of their solutions.
3. Numerical Method – the Collocation method: This chapter introduces the collocation approach, detailing the use of polynomial spline spaces and the discretization process required to solve the integral equations.
4. Order of Convergence: This section analyzes how different mesh structures, specifically uniform and asymmetrically graded meshes, influence the convergence rates of the numerical solutions.
5. Numerical examples: This chapter presents practical applications of the derived numerical methods, comparing exact solutions with approximated results through error tables.
6. References: This section lists the academic literature and foundational works used to support the mathematical frameworks and methods presented in the paper.
Keywords
Volterra integral equations, weakly singular kernels, collocation method, numerical analysis, polynomial splines, graded meshes, convergence order, product integration, quadrature formulas, integral operators, approximation theory, nonlinear equations, functional equations, computational mathematics, spline spaces.
Frequently Asked Questions
What is the primary focus of this research paper?
The paper focuses on the numerical solution of nonlinear Volterra integral equations of the second kind that possess weakly singular kernels, particularly those where the solution is non-smooth near the initial point.
What are the central themes discussed in the text?
The central themes include the classification of Volterra equations, the theoretical foundations of existence and uniqueness, the construction of collocation methods using splines, and the optimization of convergence via graded meshes.
What is the main goal or research question?
The main goal is to find an approximate solution that exhibits a high order of global convergence by employing polynomial spline approximations on specially graded meshes.
Which scientific methodology is utilized?
The paper utilizes the collocation method combined with product integration techniques and interpolatory m-point product quadrature formulas to discretize and solve the integral equations.
What does the main part of the work cover?
The main part covers the mathematical formulation of the kernel, the definition of spline spaces, the discretization of the collocation equation into a system of solvable equations, and the analysis of mesh-dependent convergence rates.
Which keywords best characterize the work?
Key terms include weakly singular Volterra integral equations, collocation method, graded meshes, convergence order, and polynomial spline spaces.
How does the choice of mesh affect the accuracy of the approximation?
Asymmetrically graded meshes are shown to be superior to uniform meshes for these specific equations because they provide a better approximation near the initial point where the kernel singularity is present.
Why is the product integration technique implemented in the discretization?
It is used because the integrals in the collocation equation cannot typically be evaluated analytically, requiring numerical quadrature to achieve the necessary precision for the approximation.
What role do Gauss-points play in the numerical examples?
Gauss-points are utilized as collocation parameters (c1, c2) in the numerical examples to facilitate the quadrature method and improve the accuracy of the spline-based approximation.
What are the potential drawbacks of using asymmetrically graded meshes mentioned by the author?
The author notes that large values of N can lead to very small mesh diameters, which might introduce rounding errors in the recursive computation process, potentially contaminating the solution.
- Citar trabajo
- Henry Ekah-Kunde (Autor), 2015, Collocation method for Weakly Singular Volterra Integral Equations of the Second Type, Múnich, GRIN Verlag, https://www.grin.com/document/370809
