Analytic solutions for singular integral equations and non-homogeneous fractional PDE

Abstract

Abstract. In the last three decades, transform methods have been used for solving fractional differential equations, singular integral equations. In this article, the author considered a new class of the inverse Laplace transforms of exponential types. We also evaluated certain types of integrals and solved partial fractional equations of Cauchy type.The result reveals that the transform method is very convenient and effective.

A Numerical Method for Weakly Singular Nonlinear Volterra Integral Equations of the Second Kind

“…Solvability and properties of singular Volterra integral equations were studied using various analytical and approximating methods. We mention existence (and uniqueness) results [1][2][3], resolvent methods [4], Laplace transforms [2,5,6], fixed point theorems [3,7], etc. Numerical solutions have been found, using product integration [8], collocation and iterated collocation [9][10][11][12], homotopy perturbation transform method [2,13], Tau method based on Jacobi functions [14], Nyström methods [8], quadrature schemes [15], variational iteration methods [6], block-pulse wavelets [16], modified quadratic spline approximation [17], reproducing kernel method [18], etc.…”