A complex fractional derivative can be derived by formally extending the integer $k$ in $k\mathrm{th}$ of a function, computed via Cauchy's integral, to $\ensuremath{\alpha}$. This straightforward approach reveals fundamental problems due inherent nonanalyticity. consequence is that not uniquely defined. We explain detail anomalies (not closed paths, branch cut jumps) and try interpret their me...

Present paper deals with the solution of time and space fractional Pennes bioheat equation. We consider time fractional derivative and space fractional derivative in the form of Caputo fractional derivative of order [Formula: see text] and Riesz-Feller fractional derivative of order [Formula: see text] respectively. We obtain solution in terms of Fox's H-function with some special cases, by usi...

In a fractional Cauchy problem, the usual first order time derivative is replaced by a fractional derivative. This problem was first considered by Nigmatullin (1986), and Zaslavsky (1994) in R for modeling some physical phenomena. The fractional derivative models time delays in a diffusion process. We will give a survey of the recent results on the fractional Cauchy problem and its generalizati...

Nonlinear time fractional partial differential equations are widely used in modeling and simulations. In many applications, there high contrast changes media properties. For solving these problems, one often uses coarse spatial grid for resolution. temporal discretization, implicit methods used. methods, though the step can be relatively large, difficult to compute due nonlinearity fact that de...

In positive systems inputs, state variables and outputs take only non-negative values. Examples of positive systems are industrial processes involving chemical reactors, heat exchangers and distillation columns, storage systems, compartmental systems, or water and atmospheric pollution models. A variety of models having positive linear behavior can be found in engineering, management science, e...

In this paper, we consider linear space-time fractional reactiondiffusion equation with composite fractional derivative as time derivative and Riesz-Feller fractional derivative with skewness zero as space derivative. We apply Laplace and Fourier transforms to obtain its solution.

In this paper, based on the fractional Riccati equation, we propose an extended fractional Riccati sub-equation method for solving fractional partial differential equations. The fractional derivative is defined in the sense of the modified Riemann-Liouville derivative. By a proposed variable transformation, certain fractional partial differential equations are turned into fractional ordinary di...

We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space ...