Please find attached a problem taken from book "Linear and Non linear Integral Equations" by Wazwaz. The given system of Volterra integral equations can be easily solved using Adomian Decomposition method, Variational Iteration method etc. I also tried solving problem using Laplace transform and i am unable to get the results. I even tried to represent transformed form as a series and then tried taking Inverse Laplace transform to get the solution but it was not fruitful. The given system is linear and Laplace transform is linear operator but i don't understand why result is not achievable. If someone can provide solution using Laplace transform then i shall be grateful.
Exact solution is u(x)=1+x2
and v(x)=1+x-x3
Regards
Saad Sharjeel
Dear Saad Sharjeel:
The integral in the first equatiuon is the Laplace convolution of x^3 and u plus the Laplace convolution of x^2 and v, and the analogouse picture in the second equation. So we should apply the Convolution Theorem for the Laplace transform to get a linear sistem withn respect to the Laplace transforms of u and v, to solve it and then apply an appropriate inersion formula.
Best regards,
Adolf.
Please study "Linear and Non-linear Integral Equations, Wazwaz". You can find many examples and useful information about your question.
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