Integral equation with unknown kernel?

In summary, the conversation discussed the problem of solving an integral equation with known functions f(x) and g(t) and a constant R. It was mentioned that this problem has infinitely many solutions, and one approach is to assume K(x,t)=F(x)G(t) and use the derivative of \frac{f(x)}{F(x)} to devise a non-linear ODE. Two specific choices of G were suggested as examples of solutions: G=\frac{1}{F} and G=\frac{1}{F^2}. The conversation also included a suggestion for a more general solution: K(x,t)=\frac{-f'(t)-\frac{f(R)}{x-R}}{g(t)}.
  • #1
benjaminmar8
10
0
Hi, all,

I would to solve an integral equation, here is the form

[tex]f(x)=\int_{x}^{R}K(x,t)g(t)dt[/tex]

f(x) and g(t) are known function, R is an constant, how to compute the unknown Kernel
K(x,t)?

Thanks a lot
 
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  • #2
Obviously such a problem has infinitely many solutions.
 
  • #3
daudaudaudau said:
Obviously such a problem has infinitely many solutions.

Could you state one for general [tex]f[/tex] and [tex]g[/tex]?What I would do first is assume we can separate [tex]K(x,t)=F(x)G(t)[/tex]
And also assume that [tex] f [/tex] and [tex] F [/tex] are differentiable.

Then you find that [tex]\frac{f(x)}{F(x)} = \int^R_x G(t)g(t)dt[/tex]
let [tex]\hat{f} = \frac{f(x)}{F(x)} [/tex]

and taking the derivative with the FTC you get that:

[tex]\hat{f}' = -G(x)g(x) [/tex]

After some calculation you can devise the non-linear ODE:

[tex]F'(x) - \frac{F^2(x)G(x)g(x)}{f(x)} - \frac{f'(x)F(x)}{f(x)} = 0[/tex]

So with specific choices for [tex]G[/tex] i.e. [tex]G=\frac{1}{F} [/tex] or[tex]G=\frac{1}{F^2}[/tex] you can get certain solutions.

for example the choice [tex]G=\frac{1}{F} [/tex] leads to the solution:

[tex] K(x,t) = e^{h(x)-h(t)}[/tex]**

where [tex] h(y)=\int^y_a \frac{g(s)+f'(s)}{f(s)}ds[/tex] for any [tex]a[/tex] in the shared domain of [tex]f,g,and\ f'[/tex]**this is actually: [tex]K(x,t)=e^{\int^x_t\frac{g(s)+f'(s)}{f(s)}ds[/tex]

which is just one solution to this problem.

edit: the solution I gave for K can be multiplied by any constant, presumably determined by [tex]R[/tex]
 
Last edited:
  • #4
WastedGunner said:
Could you state one for general [tex]f[/tex] and [tex]g[/tex]?

I think this is an obvious solution
[tex]
K(x,t)=\frac{-f'(t)-\frac{f(R)}{x-R}}{g(t)}
[/tex]

By the way, try the [ itex] [ /itex] pair when writing tex inline.
 
  • #5
Ah, I see, very simple.

And thanks for the advice.
 

1. What is an integral equation with unknown kernel?

An integral equation with unknown kernel is an equation that involves an unknown function within an integral. The unknown function is the kernel, and it is usually defined as the coefficient of the integral. Solving an integral equation with unknown kernel involves finding the unknown function that satisfies the given equation.

2. How is an integral equation with unknown kernel different from other types of equations?

An integral equation with unknown kernel is different from other types of equations because it involves an unknown function within an integral. This makes it a more complicated type of equation to solve, as it requires specific methods and techniques that are different from those used to solve other types of equations.

3. What are the applications of integral equations with unknown kernel?

Integral equations with unknown kernel have various applications in mathematics, physics, engineering, and other fields. They are commonly used to model and solve problems in heat transfer, fluid mechanics, electromagnetic theory, and signal processing, among others. They also have applications in the solution of differential equations and boundary value problems.

4. What techniques are used to solve integral equations with unknown kernel?

There are several techniques used to solve integral equations with unknown kernel, including the Fredholm method, the Neumann series method, the method of successive approximations, and the Wiener-Hopf method. Each method has its own advantages and limitations, and the choice of technique depends on the specific problem and the properties of the integral equation.

5. Are there any challenges in solving integral equations with unknown kernel?

Yes, there are some challenges in solving integral equations with unknown kernel. One of the main challenges is that there is no general method that can be applied to all types of integral equations with unknown kernel. This means that each problem must be approached individually, and the choice of technique depends on the specific properties of the equation. Additionally, these equations can also have multiple solutions, making it important to check for uniqueness in the solution.

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