How Do You Apply Convolution to Solve Integral Equations in Fourier Transforms?

[MathematicalPhysicist]

I need to find the Fourier transform of f(x) which is given by the equation:
$$-\frac{d^2f(x)}{dx^2}+\frac{1}{a^3}\int_{-\infty}^{\infty}dx'exp(-\lambda|x-x'|)f(x')=\frac{b}{a^2}exp(-\lambda|x|)$$

ofcourse Iv'e taken the Fourier tarnsform of both sides, but I don't see how to calcualte the Fourier tranform of the integral in the above equation, I feel I need to use the definition of dirac's delta function, but don't see how to do this, any ideas, hints?

thanks in advance.

 

[jhicks]

Without doing the Fourier transform, it looks to me like you'll need to use the property that $\delta(x'-x)f(x')=f(x)$. Just note which of x and x' is actually a variable for integration and which is the "constant" inside the integration. Use Parseval's Theorem.

 

[MathematicalPhysicist]

I'm not sure it's correct I got that the F.T of the integral without the constant 1/a^3 is:
2pi*f(0)*e^(a-\lambda|x|), is this correct?

 

[jhicks]

Well actually disregard my previous advice, sorry! Note that the integral is equivalent to $e^{-\lambda |x|} \ast f(x)$ where the operator is convolution. In the frequency domain, this becomes multiplication. Now you simply need to know $e^{-\lambda |x|} \Leftrightarrow \frac{2\lambda}{\lambda^2 + \omega^2}$ which is actually the same answer as you'd arrive by from what I previously said, but in much less time.

 

[MathematicalPhysicist]

so using convolution you say, ok I'll try it, thanks.