Collocation method for Weakly Singular Volterra Integral Equations of the Second Type PDF Download
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Author: Henry Ekah-Kunde
Publisher: GRIN Verlag
ISBN: 3668484260
Category : Mathematics
Languages : en
Pages : 26
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Book Description
Seminar paper from the year 2015 in the subject Mathematics - Applied Mathematics, grade: A, , language: English, abstract: 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.
Author: Henry Ekah-Kunde
Publisher: GRIN Verlag
ISBN: 3668484260
Category : Mathematics
Languages : en
Pages : 26
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Book Description
Seminar paper from the year 2015 in the subject Mathematics - Applied Mathematics, grade: A, , language: English, abstract: 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.
Author: Hermann Brunner
Publisher: Cambridge University Press
ISBN: 9780521806152
Category : Mathematics
Languages : en
Pages : 620
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Author: Herman J. J. te Riele
Publisher:
ISBN:
Category :
Languages : en
Pages : 19
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Book Description
Author: Herman H. Riele
Publisher:
ISBN:
Category :
Languages : en
Pages : 19
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Book Description
Author: Fan Bo
Publisher:
ISBN:
Category : Collocation methods
Languages : en
Pages : 190
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Book Description
Author: Fan Bai
Publisher:
ISBN:
Category : Collocation methods
Languages : en
Pages : 190
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Book Description
Author: Henry Ekah-Kunde
Publisher: GRIN Verlag
ISBN: 3668494150
Category : Mathematics
Languages : en
Pages : 23
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Book Description
Seminar paper from the year 2015 in the subject Mathematics - Applied Mathematics, grade: A, , language: English, abstract: In this research, a novel method to approximate the solution of optimal control problems governed by Volterra integral equations of weakly singular types is proposed. The method introduced here is the conjugate gradient method with a discretization of the problem based on the collocation approach on graded mesh points for non linear Volterra integral equations with singular kernels. Necessary and sufficient optimality conditions for optimal control problems are also discussed. Some examples are presented to demonstrate the efficiency of the method.
Author: L. M. Delves
Publisher: CUP Archive
ISBN: 9780521357968
Category : Mathematics
Languages : en
Pages : 392
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Book Description
This textbook provides a readable account of techniques for numerical solutions.
Author: Abdul-Majid Wazwaz
Publisher: Springer Science & Business Media
ISBN: 3642214495
Category : Mathematics
Languages : en
Pages : 639
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Book Description
Linear and Nonlinear Integral Equations: Methods and Applications is a self-contained book divided into two parts. Part I offers a comprehensive and systematic treatment of linear integral equations of the first and second kinds. The text brings together newly developed methods to reinforce and complement the existing procedures for solving linear integral equations. The Volterra integral and integro-differential equations, the Fredholm integral and integro-differential equations, the Volterra-Fredholm integral equations, singular and weakly singular integral equations, and systems of these equations, are handled in this part by using many different computational schemes. Selected worked-through examples and exercises will guide readers through the text. Part II provides an extensive exposition on the nonlinear integral equations and their varied applications, presenting in an accessible manner a systematic treatment of ill-posed Fredholm problems, bifurcation points, and singular points. Selected applications are also investigated by using the powerful Padé approximants. This book is intended for scholars and researchers in the fields of physics, applied mathematics and engineering. It can also be used as a text for advanced undergraduate and graduate students in applied mathematics, science and engineering, and related fields. Dr. Abdul-Majid Wazwaz is a Professor of Mathematics at Saint Xavier University in Chicago, Illinois, USA.
Author: Scotty Glen Houston
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
We implement several methods for solving Volterra integral equations based on standard techniques for solving ordinary differential equations. In particular we implement a version of the modified Euler method,we implement the standard fourth order Runge-Kutta method.We compare these with a novel collocation method using Bernstein polynomials.We also look at Volterra integral equations with weakly singular kernels. Here the standard methods developed above do not apply though the collocation method can still be used. We develop two “product integration” methods, derived from numerical integration methods that can be applied to these weakly singular Volterra integral equations and compare them to the collocation method.
