A certain Fredholm partial integro-differential equation of the third order

We study the unique solvability of the initial value problem for a nonlinear partial Fredholm integro-differential equation of the third order with a degenerate kernel. First, we adapt the degenerate kernel method developed for a partial Fredholm integro-differential equation of the second kind for solving Fredholm integro-differential equations of the third order. After solving the corresponding system of algebraic equations, we obtain a Volterra integral equation of the second kind. Then we apply the method of successive approximations combined with the method of contractive maps.

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