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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: 1.1 $$u\left( x \right) = f\left( x \right) + \int_{g\left( x \right)}^{h\left( x \right)} {K\left( {x,t} \right)u\left( t \right)dt,} $$ where g(x) and h(x) are the limits of integration, λ is a constant parameter, and K(x, t)...
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 2.1 $$u\left( x \right) = f\left( x \right) + \int_{g\left( x \right)}^{h\left( x \right)} {K\left( {x,t} \right)u\left( t \right)dt,} $$ where g(x) and h(x) are the limits of integration,...
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...
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...
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 5.1 $${u^{\left( n \right)}}\left( x \right) = f\left( x \right)...
In Chapter 2, the conversion of boundary value problems to Fredholm integral equations was presented. However, the research work in this field resulted in a new specific topic, where both differential and integral operators appeared together in the same equation. This new type of equations, with constant limits of integration, was termed as Fredholm integro-differential equations, given in the form...
Abel’s integral equation occurs in many branches of scientific fields [1], such as microscopy, seismology, radio astronomy, electron emission, atomic scattering, radar ranging, plasma diagnostics, X-ray radiography, and optical fiber evaluation. Abel’s integral equation is the earliest example of an integral equation [2]. In Chapter 2, Abel’s integral equation was defined as a singular integral equation...
The Volterra-Fredholm integral equations [1–2] arise from parabolic boundary value problems, from the mathematical modelling of the spatio-temporal development of an epidemic, and from various physical and biological models. The Volterra-Fredholm integral equations appear in the literature in two forms, namely 8.1 $$u\left( x \right) = f\left( x \right) = {\lambda _1}\int_0^x {{K_1}\left( {x,t}...
The Volterra-Fredholm integro-differential equations [1–4] appear in two types, namely: 9.1 $${u^{\left( k \right)}}\left( x \right) = f\left( x \right) + {\lambda _1}\int_a^x {{K_1}\left( {x,t} \right)u\left( t \right)dt + {\lambda _2}\int_a^b {{K_2}\left( {x,t} \right)u\left( t \right)dt} ,} $$ and the mixed form 9.2 $${u^{\left( k \right)}}\left( x \right) = f\left( x \right) + \lambda \int_0^x...
Systems of integral equations, linear or nonlinear, appear in scientific applications in engineering, physics, chemistry and populations growth models [1–4]. Studies of systems of integral equations have attracted much concern in applied sciences. The general ideas and the essential features of these systems are of wide applicability.
Systems of Volterra and Fredholm integral equations have attracted much concern in applied sciences. The systems of Fredholm integral equations appear in two kinds. The system of Fredholm integral equations of the first kind [1–5] reads 11.1 $$\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)}...
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...
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...
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.
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...
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.
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