Which of the following is a Fredholm integral equation?

Consider the following Fredholm integral equation of second kind:(1) u ( x ) = f ( x ) + λ ∫ a b k ( x , t ) F ( u ( t ) ) dt , x , t ∈ [ a , b ] , where λ is a real number, also F, f and k are given continuous functions, and u is unknown function to be determined.

What is Fredholm integral equation of the first kind?

Moreover, Fredholm integral equations of the first kind are of the form (2) f ( x ) = λ ∫ a b K ( x , t ) u ( t ) d t , x ∈ Ω , where is a closed and bounded region. Fredholm integral equations of the first kind (2) are characterized by the occurrence of the unknown function only inside the integral sign.

How many types of Fredholm integral equations are there?

There are four basic types of integral equations. There are many other integral equations, but if you are familiar with these four, you have a good overview of the classical theory. All four involve the unknown function φ(x) in an integral with a kernel K(x, y) and all have an input function f(x).

Who invented Volterra integral equation?

Vito Volterra
The Volterra integral equations were introduced by Vito Volterra and then studied by Traian Lalescu in his 1908 thesis, Sur les équations de Volterra, written under the direction of Émile Picard. In 1911, Lalescu wrote the first book ever on integral equations.

What is integral form?

The integral form of the full equations is a macroscopic statement of the principles of conservation of mass and momentum for what is called a control volume. A control volume is a conceptual device for clearly describing the various fluxes and forces in open-channel flow.

What is Volterra integro differential equation?

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.

Which is linear integral equation?

INTEGRAL EQUATIONS In this equation the function ϕ is the unknown. The equation is a linear integral equation because ϕ appears in a linear form (i.e., we do not have terms like ϕ2). If a = 0 then we have a Fredholm integral equation of the first kind. In these equations the unknown appears only in the integral term.

What is integral equation theory?

Liquid state integral equation theory was originally developed for atomic and small molecule fluids, but has in the last decades found widespread applications in colloids science. The theory provides a (inter-particle) pair correlation function when the inter-particle potential is specified.

What do you mean by integral equation?

integral equation, in mathematics, equation in which the unknown function to be found lies within an integral sign. An example of an integral equation is. in which f(x) is known; if f(x) = f(-x) for all x, one solution is.

What is Volterra integral equation of second kind?

In mathematics, the Volterra integral equations are a special type of integral equations. They are divided into two groups referred to as the first and the second kind. In operator theory, and in Fredholm theory, the corresponding operators are called Volterra operators.

What is homogeneous integral equation?

[‚hä·mə′jē·nē·əs ′int·ə·grəl i‚kwā·zhən] (mathematics) An integral equation where every scalar multiple of a solution is also a solution.

What is integral of DT?

The integration of dt is (t+c). If a function F(x) is found such that the differentiation of F(x) equals f(x), then F(x) is called the integral of f(x).