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2011 | Buch

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Linear and Nonlinear Integral Equations

Methods and Applications

verfasst von: Abdul-Majid Wazwaz

Verlag: Springer Berlin Heidelberg

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Linear Integral Equations

Frontmatter

Chapter 1. Preliminaries

Abstract

An integral equation is an equation in which the unknown function u(x) appears under an integral sign [1&#sx20s13;7]. A standard integral equation in u(x) is of the form:

$$u\left( x \right) = f\left( x \right) + \int_{g\left( x \right)}^{h\left( x \right)} {K\left( {x,t} \right)u\left( t \right)dt,} $$

(1.1)

where g(x) and h(x) are the limits of integration, λ is a constant parameter, and K(x, t) is a function of two variables x and t called the kernel or the nucleus of the integral equation. The function u(x) that will be determined appears under the integral sign, and it appears inside the integral sign and outside the integral sign as well. The functions f(x) and K(x, t) are given in advance. It is to be noted that the limits of integration g(x) and h(x) may be both variables, constants, or mixed.

Abdul-Majid Wazwaz

Chapter 2. Introductory Concepts of Integral Equations

Abstract

As stated in the previous chapter, an integral equation is the equation in which the unknown function u(x) appears inside an integral sign [1–5]. The most standard type of integral equation in u(x) is of the form

$$u\left( x \right) = f\left( x \right) + \int_{g\left( x \right)}^{h\left( x \right)} {K\left( {x,t} \right)u\left( t \right)dt,} $$

(2.1)

where g(x) and h(x) are the limits of integration, λ is a constant parameter, and K(x, t) is a known function, of two variables x and t, called the kernel or the nucleus of the integral equation. The unknown function u(x) that will be determined appears inside the integral sign. In many other cases, the unknown function u(x) appears inside and outside the integral sign. The functions f(x) and K(x, t) are given in advance. It is to be noted that the limits of integration g(x) and h(x) may be both variables, constants, or mixed.

Abdul-Majid Wazwaz

Chapter 3. Volterra Integral Equations

Abstract

It was stated in Chapter 2 that Volterra integral equations arise in many scientific applications such as the population dynamics, spread of epidemics, and semi-conductor devices. It was also shown that Volterra integral equations can be derived from initial value problems. Volterra started working on integral equations in 1884, but his serious study began in 1896. The name sintegral equation was given by du Bois-Reymond in 1888. However, the name Volterra integral equation was first coined by Lalesco in 1908.

Abdul-Majid Wazwaz

Chapter 4. Fredholm Integral Equations

Abstract

It was stated in Chapter 2 that Fredholm integral equations arise in many scientific applications. It was also shown that Fredholm integral equations can be derived from boundary value problems. Erik Ivar Fredholm (1866– 1927) is best remembered for his work on integral equations and spectral theory. Fredholm was a Swedish mathematician who established the theory of integral equations and his 1903 paper in Acta Mathematica played a major role in the establishment of operator theory.

Abdul-Majid Wazwaz

Chapter 5. Volterra Integro-Differential Equations

Abstract

Volterra studied the hereditary influences when he was examining a population growth model. The research work resulted in a specific topic, where both differential and integral operators appeared together in the same equation. This new type of equations was termed as Volterra integro-differential equations [1–4], given in the form

$${u^{\left( n \right)}}\left( x \right) = f\left( x \right) + \lambda \int_0^{\left( x \right)} {K\left( {x,t} \right)u\left( t \right)dt,} $$

(5.1)

Where ${u^{\left( n \right)}}\left( x \right) = \frac{{{d^n}u}}{{d{x^n}}}$. Because the resulted equation in (5.1) combines the differential operator and the integral operator, then it is necessary to define initial conditions u(0), u′ (0), , u ⁽ⁿ⁻¹⁾(0) for the determination of the particular solution u(x) of the Volterra integro-differential equation (5.1). Any Volterra integro-differential equation is characterized by the existence of one or more of the derivatives u′ (x), u″ (x), outside the integral sign. The Volterra integro-differential equations may be observed when we convert an initial value problem to an integral equation by using Leibnitz rule.

Abdul-Majid Wazwaz

Chapter 6. Fredholm Integro-Differential Equations

Abstract

In Chapter 2, the conversion of boundary value problems to Fredholm integral equations was presented. However, the research work in this field resulted in a new specific topic, where both differential and integral operators appeared together in the same equation. This new type of equations, with constant limits of integration, was termed as Fredholm integro-differential equations, given in the form

$${u^{\left( n \right)}}\left( x \right) = f\left( x \right) + \int_a^b {K\left( {x,t} \right)u\left( t \right)dt,{u^{\left( k \right)}}\left( 0 \right) = {b_k},0 \leqslant k \leqslant n - 1,} $$

(6.1)

where ${u^{\left( n \right)}}\left( x \right) = \frac{{{d^n}u}}{{d{x^n}}}$. Because the resulted equation in (6.1) combines the differential operator and the integral operator, then it is necessary to define initial conditions u(0), u′ (0), , u ⁽ⁿ⁻¹⁾(0) for the determination of the particular solution u(x) of equation (6.1). Any Fredholm integro-differential equation is characterized by the existence of one or more of the derivatives u′, (x), u″(x), outside the integral sign. The Fredholm integro-differential equations of the second kind appear in a variety of scientific applications such as the theory of signal processing and neural networks [1–3].

Abdul-Majid Wazwaz

Chapter 7. Abel’s Integral Equation and Singular Integral Equations

Abstract

Abel’s integral equation occurs in many branches of scientific fields [1], such as microscopy, seismology, radio astronomy, electron emission, atomic scattering, radar ranging, plasma diagnostics, X-ray radiography, and optical fiber evaluation. Abel’s integral equation is the earliest example of an integral equation [2]. In Chapter 2, Abel’s integral equation was defined as a singular integral equation. Volterra integral equations of the first kind

$$f\left( x \right) = \lambda \int_{g\left( x \right)}^{h\left( x \right)} {K\left( {x,t} \right)u\left( t \right)dt,} $$

(7.1)

or of the second kind

$$u\left( x \right) = f\left( x \right) = \lambda \int_{g\left( x \right)}^{h\left( x \right)} {K\left( {x,t} \right)u\left( t \right)dt,} $$

(7.2)

are called singular [3–4] if:

one of the limits of integration g(x), h(x) or both are infinite, or

if the kernel K(x, t) becomes infinite at one or more points at the range of integration.

Abdul-Majid Wazwaz

Chapter 8. Volterra-Fredholm Integral Equations

Abstract

The Volterra-Fredholm integral equations [1–2] arise from parabolic boundary value problems, from the mathematical modelling of the spatio-temporal development of an epidemic, and from various physical and biological models. The Volterra-Fredholm integral equations appear in the literature in two forms, namely

$$u\left( x \right) = f\left( x \right) = {\lambda _1}\int_0^x {{K_1}\left( {x,t} \right)u\left( t \right)dt + {\lambda _2}} \int_a^b {{K_2}\left( {x,t} \right)u\left( t \right)dt} ,$$

(8.1)

and the mixed form

$$u\left( x \right) = f\left( x \right) = \lambda \int_0^x {\int_a^b {K\left( {r,t} \right)u\left( t \right)dtdr} ,} $$

(8.2)

where f(x) and K(x, t) are analytic functions. It is interesting to note that (8.1) contains disjoint Volterra and Fredholm integrals, whereas (8.2) contains mixed Volterra and Fredholm integrals. Moreover, the unknown functions u(x) appears inside and outside the integral signs. This is a characteristic feature of a second kind integral equation. If the unknown functions appear only inside the integral signs, the resulting equations are of first kind. Examples of the two types of the Volterra-Fredholm integral equations of the second kind are given by

$$u\left( x \right) = 6x + 3{x^2} + 2 - \int_0^x {xu\left( t \right)dt - \int_0^1 {tu\left( t \right)dt} ,} $$

(8.3)

and

$$u\left( x \right) = x + \frac{{17}}{2}{x^2} - \int_0^x {\int_0^1 {\left( {r - t} \right)u\left( t \right)drdt} .} $$

(8.4)

Abdul-Majid Wazwaz

Chapter 9. Volterra-Fredholm Integro-Differential Equations

Abstract

The Volterra-Fredholm integro-differential equations [1–4] appear in two types, namely:

$${u^{\left( k \right)}}\left( x \right) = f\left( x \right) + {\lambda _1}\int_a^x {{K_1}\left( {x,t} \right)u\left( t \right)dt + {\lambda _2}\int_a^b {{K_2}\left( {x,t} \right)u\left( t \right)dt} ,} $$

(9.1)

and the mixed form

$${u^{\left( k \right)}}\left( x \right) = f\left( x \right) + \lambda \int_0^x {\int_a^b {K\left( {r,t} \right)u\left( t \right)dtdr} ,} $$

(9.2)

where ${u^{\left( k \right)}}\left( x \right) = \frac{{{d^k}u\left( x \right)}}{{d{x^k}}}$. The first type contains disjoint integrals and the second type contains mixed integrals such that the Fredholm integral is the interior one, and Volterra is the exterior integral.

Abdul-Majid Wazwaz

Chapter 10. Systems of Volterra Integral Equations

Abstract

Systems of integral equations, linear or nonlinear, appear in scientific applications in engineering, physics, chemistry and populations growth models [1–4]. Studies of systems of integral equations have attracted much concern in applied sciences. The general ideas and the essential features of these systems are of wide applicability.

Abdul-Majid Wazwaz

Chapter 11. Systems of Fredholm Integral Equations

Abstract

Systems of Volterra and Fredholm integral equations have attracted much concern in applied sciences. The systems of Fredholm integral equations appear in two kinds. The system of Fredholm integral equations of the first kind [1–5] reads

$$\begin{gathered} {f_1}\left( x \right) = \int_a^b {\left( {{K_1}\left( {x,t} \right)u\left( t \right) + {{\tilde K}_1}\left( {x,t} \right)v\left( t \right)} \right)} dt, \hfill \\ {f_2}\left( x \right) = \int_a^b {\left( {{K_2}\left( {x,t} \right)u\left( t \right) + {{\tilde K}_2}\left( {x,t} \right)v\left( t \right)} \right)} dt, \hfill \\ \end{gathered} $$

(11.1)

where the unknown functions u(x) and v(x) appear only under the integral sign, and a and b are constants. However, for systems of Fredholm integral equations of the second kind, the unknown functions u(x) and v(x) appear inside and outside the integral sign. The second kind is represented by the form

$$\begin{gathered} u\left( x \right) = {f_1}\left( x \right) + \int_a^b {\left( {{K_1}\left( {x,t} \right)u\left( t \right) + {{\tilde K}_1}\left( {x,t} \right)v\left( t \right)} \right)} dt, \hfill \\ v\left( x \right) = {f_2}\left( x \right) + \int_a^b {\left( {{K_2}\left( {x,t} \right)u\left( t \right) + {{\tilde K}_2}\left( {x,t} \right)v\left( t \right)} \right)} dt. \hfill \\ \end{gathered} $$

(11.2)

The systems of Fredholm integro-differential equations have also attracted a considerable size of interest. These systems are given by

$$\begin{gathered} {u^{\left( i \right)}}\left( x \right) = {f_1}\left( x \right) + \int_a^b {\left( {{K_1}\left( {x,t} \right)u\left( t \right) + {{\tilde K}_1}\left( {x,t} \right)v\left( t \right)} \right)} dt, \hfill \\ {v^{\left( i \right)}}\left( x \right) = {f_2}\left( x \right) + \int_a^b {\left( {{K_2}\left( {x,t} \right)u\left( t \right) + {{\tilde K}_2}\left( {x,t} \right)v\left( t \right)} \right)} dt, \hfill \\ \end{gathered} $$

(11.3)

where the initial conditions for the last system should be prescribed.

Abdul-Majid Wazwaz

Chapter 12. Systems of Singular Integral Equations

Abstract

Systems of singular integral equations appear in many branches of scientific fields [1–6], such as microscopy, seismology, radio astronomy, electron emission, atomic scattering, radar ranging, plasma diagnostics, X-ray radiography, and optical fiber evaluation. Studies of systems of singular integral equations have attracted much concern in applied sciences. The use of computer symbolic systems such as Maple and Mathematica facilitates the tedious work of computation. The general ideas and the essential features of these systems are of wide applicability.

Abdul-Majid Wazwaz

Nonlinear Integral Equations

Frontmatter

Chapter 13. Nonlinear Volterra Integral Equations

Abstract

It is well known that linear and nonlinear Volterra integral equations arise in many scientific fields such as the population dynamics, spread of epidemics, and semi-conductor devices. Volterra started working on integral equations in 1884, but his serious study began in 1896. The name integral equation was given by du Bois-Reymond in 1888. However, the name Volterra integral equation was first coined by Lalesco in 1908.

Abdul-Majid Wazwaz

Chapter 14. Nonlinear Volterra Integro-Differential Equations

Abstract

Abdul-Majid Wazwaz

Chapter 15. Nonlinear Fredholm Integral Equations

Abstract

It was stated in Chapter 4 that Fredholm integral equations arise in many scientific applications. It was also shown that Fredholm integral equations can be derived from boundary value problems. Erik Ivar Fredholm (1866–1927) is best remembered for his work on integral equations and spectral theory. Fredholm was a Swedish mathematician who established the theory of integral equations and his 1903 paper in Acta Mathematica played a major role in the establishment of operator theory. The linear Fredholm integral equations and the linear Fredholm integro-differential equations were presented in Chapters 4 and 6 respectively. It is our goal in this chapter to study the nonlinear Fredholm integral equations of the second kind and systems of nonlinear Fredholm integral equations of the second kind.

Abdul-Majid Wazwaz

Chapter 16. Nonlinear Fredholm Integro-Differential Equations

Abstract

The linear Fredholm integral equations and the linear Fredholm integrodifferential equations were presented in Chapters 4 and 6 respectively. In Chapter 15, the nonlinear Fredholm integral equations were examined. It is our goal in this chapter to study the nonlinear Fredholm integro-differential equations [1–7] and the systems of nonlinear Fredholm integro-differential equations.

Abdul-Majid Wazwaz

Chapter 17. Nonlinear Singular Integral Equations

Abstract

Abel’s integral equation, linear or nonlinear, occurs in many branches of scientific fields [1], such as microscopy, seismology, radio astronomy, electron emission, atomic scattering, radar ranging, plasma diagnostics, X-ray radiography, and optical fiber evaluation. Linear Abel’s integral equation is the earliest example of an integral equation. In Chapter 2, Abel’s integral equation was defined as a singular integral equation. Volterra integral equations of the first kind

$$f\left( x \right) = \lambda \int_{g\left( x \right)}^{h\left( x \right)} {K\left( {x,t} \right)u\left( t \right)dt,} $$

(17.1)

or of the second kind

$$u\left( x \right) = f\left( x \right) + \int_{g\left( x \right)}^{h\left( x \right)} {K\left( {x,t} \right)u\left( t \right)dt,} $$

(17.2)

are called singular [2–8] if:

one of the limits of integration g(x), h(x) or both are infinite, or

if the kernel K(x, t) becomes infinite at one or more points at the range of integration.

Abdul-Majid Wazwaz

Chapter 18. Applications of Integral Equations

Abstract

Integral equations arise in many scientific and engineering problems. A large class of initial and boundary value problems can be converted to Volterra or Fredholm integral equations. The potential theory contributed more than any field to give rise to integral equations. Mathematical physics models, such as diffraction problems, scattering in quantum mechanics, conformal mapping, and water waves also contributed to the creation of integral equations.

Abdul-Majid Wazwaz

Backmatter

Titel: Linear and Nonlinear Integral Equations
verfasst von: Abdul-Majid Wazwaz
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-21449-3
Print ISBN: 978-3-642-21448-6
DOI: https://doi.org/10.1007/978-3-642-21449-3