This survey is devoted to some fractional extensions of the incomplete lumped formulation, the lumped formulation and the formulation of Lauwerier of the temperature field problem in oil strata. The method of integral transforms is used to solve the corresponding boundary value problems for the fractional heat equation. By using Caputo’s differintegration operator and the Laplace transform, new integral forms of the solutions are obtained. In each of the different cases the integrands are expressed...

Mathematical Subject Classification 2010:26A33, 33E99, 15A52, 62E15.Mittag-Leffler functions and their generalizations appear in a large variety of problems in different areas. When we move from total differential equations to fractional equations Mittag-Leffler functions come in naturally. Fractional reaction-diffusion problems in physical sciences and general input-output models in other disciplines are some of the examples in this direction. Some basic properties of Mittag-Leffler functions are...

In this paper, we consider a fractional impulsive boundary value problem on infinite intervals. We obtain the existence, uniqueness and computational method of unbounded positive solutions.

We study the boundedness of the one-sided operator $g{⁺}_{\lambda ,\phi }$ between the weighted spaces $L^p\left(M¯w\right)$ and $L^p\left(w\right)$ for every weight w. If λ = 2/p whenever 1 < p < 2, and in the case p = 1 for λ > 2, we prove the weak type of $g{⁺}_{\lambda ,\phi }$. For every λ > 1 and p = 2, or λ > 2/p and 1 < p < 2, the boundedness of this operator is obtained. For p > 2 and λ > 1, we obtain the boundedness of $g{⁺}_{\lambda ,\phi }$ from $L^p\left({\left(M¯\right)}^{[p/2]+1}w\right)$ to $L^p\left(w\right)$, where ${\left(M¯\right)}^k$ denotes the operator M¯ iterated k times.

We study a stochastic fractional partial differential equations of order $\alpha \&gt;1$ driven by a compensated Poisson measure. We prove existence and uniqueness of the solution and we study the regularity of its trajectories.

Mathematics Subject Classification: 26A33, 76M35, 82B31A stochastic solution is constructed for a fractional generalization of the KPP (Kolmogorov, Petrovskii, Piskunov) equation. The solution uses a fractional generalization of the branching exponential process and propagation processes which are spectral integrals of Levy processes.

MSC 2010: 26A33, 34A08, 34K37

In this article, we present a new method for converting the boundary value problems for impulsive fractional differential systems involved with the Riemann-Liouville type derivatives to integral systems, some existence results for solutions of a class of boundary value problems for nonlinear impulsive fractional differential systems at resonance case and non-resonance case are established respectively. Our analysis relies on the well known Schauder’s fixed point theorem and coincidence degree theory....

Mathematics Subject Classification: 26A33 (main), 35A22, 78A25, 93A30The generalization of the concept of derivative to non-integer values goes back to the beginning of the theory of differential calculus. Nevertheless, its application in physics and engineering remained unexplored up to the last two decades. Recent research motivated the establishment of strategies taking advantage of the Fractional Calculus (FC) in the modeling and control of many phenomena. In fact, many classical engineering...

We present a new method to study the classical fractional integrals of Weyl. This new approach basically consists in considering these operators in the largest space where they make sense. In particular, we construct a theory of fractional integrals of Weyl by studying these operators in an appropriate Fréchet space. This is a function space which contains the $L^p\left(ℝ\right)$-spaces, and it appears in a natural way if we wish to identify these fractional operators with fractional powers of a suitable non-negative...

We consider the summation equation, for $t\in {[\mu -2,\mu +b]}_{{\mathbb{N}}_{\mu -2}}$, $$\begin{array}{ccc}\hfill y\left(t\right)={\gamma }_1\left(t\right)H_1\left(\sum _{i=1}^n{a}_{i}y\left({\xi }_i\right)\right)& +{\gamma }_2\left(t\right)H_2\left(\sum _{i=1}^m{b}_{i}y\left({\zeta }_i\right)\right)\hfill & \hfill +\lambda \sum _{s=0}^{b}G(t,s)f(s+\mu -1,y(s+\mu -1))\end{array}$$ in the case where the map $(t,s)\mapsto G(t,s)$ may change sign; here $\mu \in (1,2]$ is a parameter, which may be understood as the order of an associated discrete fractional boundary value problem. In spite of the fact that $G$ is allowed to change sign, by introducing a new cone we are able to establish the existence of at least one positive solution to this problem by imposing some growth conditions on the functions $H_1$ and $H_2$. Finally, as an application of the abstract existence result,...