Editing Integral equation (section)

== Volterra integral equations ==

=== Uniqueness and existence theorems in 1D ===
The solution to a linear Volterra integral equation of the first kind, given by the equation:<math display="block">(\mathcal{V}y)(t)=g(t)</math>can be described by the following uniqueness and existence theorem.<ref name=":2" /> Recall that the Volterra integral operator <math>\mathcal{V} : C(I) \to C(I)</math>, can be defined as follows:<ref name=":2" /><math display="block">(\mathcal{V} \varphi)(t) := \int_{t_0}^t K(t,s) \, \varphi(s) \, ds</math>where <math>t \in I = [t_0 , T]</math> and ''K''(''t'',''s'') is called the kernel and must be continuous on the interval <math>D := \{(t,s) : 0 \leq s \leq t \leq T \leq \infty\}</math>.<ref name=":2" /> 
{{Math theorem
| math_statement = Assume that <math> K </math> satisfies <math> K \in C(D), \, \partial K / \partial t \in C(D) </math> and <math> \vert K(t,t) \vert  \geq k_0 > 0 </math> for some <math> t \in I. </math> Then for any <math> g\in C^1(I) </math> with <math> g(0)=0 </math> the integral equation above has a unique solution in <math> y \in C(I)</math>.
}}
The solution to a linear Volterra integral equation of the second kind, given by the equation:<ref name=":2" /><math display="block">y(t)=g(t)+(\mathcal{V} y)(t)</math>can be described by the following uniqueness and existence theorem.<ref name=":2" /> 
{{Math theorem
| math_statement = Let <math> K \in C(D) </math> and let <math>R </math> denote the resolvent Kernel associated with <math> K </math>. Then, for any <math>g \in C(I) </math>, the second-kind Volterra integral equation has a unique solution  <math>y \in C(I) </math> and this solution is given by: <math>y(t)=g(t)+\int_0^t R(t,s) \, g(s) \, ds.</math>
}}

=== Volterra integral equations in {{math|R{{sup|2}}}}===

A Volterra Integral equation of the second kind can be expressed as follows:<ref name=":2" /><math display="block">u(t,x) = g(t,x)+\int_0^x \int_0^y K(x,\xi, y, \eta) \, u(\xi, \eta) \, d\eta \, d\xi</math>where <math>(x,y) \in \Omega := [0,X] \times [0,Y]</math>, <math>g \in C( \Omega)</math>, <math>K \in C(D_2)</math> and <math>D_2 := \{(x, \xi,y,\eta): 0 \leq \xi \leq x \leq X, 0 \leq \eta \leq y \leq Y\}</math>.<ref name=":2" /> This integral equation has a unique solution <math>u \in C( \Omega)</math> given by:<ref name=":2" /><math display="block">u(t,x) = g(t,x)+\int_0^x \int_0^{y} R(x,\xi, y, \eta) \, g(\xi, \eta) \, d\eta \, d\xi</math>where <math>R</math> is the resolvent kernel of ''K''.<ref name=":2" />

=== Uniqueness and existence theorems of Fredholm–Volterra equations ===
As defined above, a VFIE has the form:<math display="block">u(t,x) = g(t,x)+(\mathcal{T}u)(t,x)</math>with <math>x \in \Omega</math> and <math>\Omega</math> being a closed bounded region in <math>\mathbb{R}^d</math> with piecewise smooth boundary.<ref name=":2" /> The Fredholm–Volterrra Integral Operator <math>\mathcal{T} : C(I \times \Omega) \to C(I \times \Omega)</math> is defined as:<ref name=":2" /><math display="block">(\mathcal{T}u)(t,x) := \int_0^t \int_\Omega K(t,s,x,\xi) \, G(u(s, \xi)) \, d\xi \, ds.</math>In the case where the Kernel ''K'' may be written as <math>K(t,s,x,\xi) = k(t-s)H(x, \xi)</math>, ''K'' is called the positive memory kernel.<ref name=":2" /> With this in mind, we can now introduce the following theorem:<ref name=":2" />
{{Math theorem
| math_statement = If the linear VFIE given by: <math> u(t,x) = g(t,x) + \int_0^t \int_{\Omega} K(t,s,x,\xi) \, G(u(s, \xi)) \, d\xi \, ds </math>  with  <math> (t,x) \in I \times \Omega </math> satisfies the following conditions:
* <math>g \in C(I \times \Omega)</math>, and
* <math> K \in C(D \times \Omega^2) </math> where  <math>  D:= \{(t,s): 0 \leq s \leq t \leq T \}  </math>  and  <math> \Omega^2 = \Omega \times \Omega</math>

Then the VFIE has a unique solution <math> u \in C(I \times \Omega) </math> given by <math> u(t,x) = g(t,x)+\int_0^t \int_{\Omega} R(t,s,x,\xi) \, G(u(s, \xi)) \, d\xi \, ds </math> where <math> R \in C(D \times \Omega^2) </math> is called the Resolvent Kernel and is given by the limit of the Neumann series for the Kernel <math> K  </math> and solves the resolvent equations: <math> R(t,s,x,\xi) = K(t,s,x,\xi)+\int_0^t \int_\Omega K(t,v,x,z) R(v,s,z,\xi) \, dz \, dv =  K(t,s,x,\xi)+\int_0^t \int_\Omega R(t,v,x,z) K(v,s,z,\xi) \, dz \, dv  </math>
}}

=== Special Volterra equations ===
A special type of Volterra equation which is used in various applications is defined as follows:<ref name=":2" /><math display="block">y(t)=g(t)+(V_\alpha y)(t)</math>where <math>t \in I = [t_0 , T]</math>, the function ''g''(''t'') is continuous on the interval <math>I</math>, and the Volterra integral operator <math>(V_\alpha t)</math> is given by:<math display="block">(V_\alpha t)(t) := \int_{t_0}^t (t-s)^{-\alpha} \cdot k(t,s,y(s)) \, ds </math>with <math>(0 \leq \alpha < 1)</math>.<ref name=":2" />