Solution to time fractional non homogeneous first order PDE with non constant coefficients

Direct methods for singular integral equations and non-homogeneous parabolic PDEs

“…We recall a useful lemma about the L 2 -transform of the δ-derivatives. 1) are all continuous and of exponential order exp(c 2 t 2 ) as t → ∞ for some real constant c and piecewise continuous derivative f (n) on the interval t ≥ 0 (1) For n = 1, 2, . .…”

“…Definition 1.1. The Laplace transform of function f (t) is given by [1,4,6] L{f (t); s} = ∞ 0 e −st f (t)dt = F (s). (1.1) If L{f (t); s} = F (s), then L −1 {F (s)} is as follows…”

“…Proof. See [1,6]. □ Singular integral equations arise frequently in the mathematical modeling of continuum phenomena, and many a time cannot be treated by known analytical techniques.…”

Solution to unsteady fractional heat conduction in the quarter-plane via the joint Laplace-Fourier sine transforms

“…We provided methods and results for a partial fractional differential equations which arise in applications. Different methods of solution have been introduced to solve partial fractional differential equations, the Laplace transform method, [1], [2], [3], the Fourier transform method [10], operational method [4], [6]. The diffusion equation describes the flow of heat, or a concentration of particles.…”