Partial Integro-Differential Equations.

Integro differential equation has been used for the mathematical modeling of various fields such as engineering sciences and physical, biological phenomena.

An integro-differential equation (IDE) is explained as an equationcontaining unknown function f(x), with both differential and integral operationson f4&6

In ordinary integro- differential equation the derivative is taken with respect to single variable.In case of partial integro- differential equations derivatives is with respect to different variables.

The different types of integro- differentialequations is explain in followingsections.

Ordinary integro- differentialequations(OIDEs)

Linear ordinary integro- differentialequations:

This type of equation is represented as

h(x) ?f/?x=g(x)+??_a^(b(x))??k(x,y)f(y)dy….?

Where the known function of x, are h and g.The kernel of this equation is denoted by k(x,y) which is a known function of x andy.The known parameters are a,? and f is the unknown function.

Next , we classified two types of the linear (OIDEs), namely Fredholm and Voltera types.

Fredholm linear ordinary integro- differentialequations:

In equation ………..if upper limit is not function of x ieif b(x)=b then eq.(1) is termed Fredholm linear OIDEs. For h(x)=0 then equation 1 is called the first kind

If the limit of the integral operator in eq. (1) does not depends on x i.e. if b(x)=b then eq.(1) is called Fredholm linear OIDEs. In this case if h(x)=0 then eq.(1)reduces to the following equation:

g(x)=?_a^b??k(x,y)f(y)dy?

In equation 1 if h(x)=1then this equation is called as second kind Fredholm linear OIDEs order as given

h(x) ?f/?x=g(x)+??_a^b??k(x,y)f(y)dy?

Ifh and if g(x)=0 then eq.(1) reduces

h(x) ?f/?x=??_a^b??k(x,y)f(y)dy?

Which is termedas the third kind Fredholm linear OIDEs

Volterra linear ordinary integro- differentialequations:

The equation 1 becomes Volterra linear OIDEs.if b(x)=x i.e

h(x) ?f/?x=g(x)+??_a^x??k(x,y)f(y)dy?

and similar to the Fredholm linear OIDEs, the Volterra linear OIDEs can be categorized into first, second and third kinds.

1.2Nonlinear ordinary integro- differential equations:

Generally the nonlinear OIDE is written as:

h(x) ?f/?x=g(x)+??_a^(b(x))??k(x,y)f(y)dy….?

Similar to linear ordinary integro- differentialequations the nonlinearordinary integro- differentialequationscan be categorized into Fredholm, Volterra of the different kinds i.e first, second and third.

Partial integro- differentialequation(PIDE)

This equation is second type of an integro- differential equation in which

Unknown function depends on more than one independent variablelike the OIDEs, the PIDEs is classified into linear andnonlinear.

Linear partial integro- differentialequations:

The linear PIDE is represented as

h(x,y) (?f(x,y))/?x=g(x,y)+??_a^(b(x))??_c^(d(y))??k(x,y,z,m)f(z,m)dzdm……3?

The equation 3 is termed as Fredholm linearpartial integro- differentialequations if integral limit b(x)&d(y) are not depend onx and y.The equation 3 become Volterra linear PIDE if b(x)=x and d(y)=y

Nonlinear partial integro- differentialequations:

The nonlinear partial integro-differential equations is written as

h(x,y) (?f(x,y))/?x=g(x,y)+??_a^(b(x))??_c^(d(y))??k(x,y,z,m)f(z,m)dzdm……4?

And similar to the linear PIDEs, the nonlinear PIDEs can be devided into Volterra ,Fredholm of the first, second and third kinds.