Fractional Cauchy Research Articles

Stochastic solutions of conformable fractional Cauchy problems

In this paper we give stochastic solutions of conformable fractional Cauchy problems. The stochastic solutions are obtained by running the processes corresponding to Cauchy problems with nonlinear deterministic clocks.

New Result on the Critical Exponent for Solution of an Ordinary Fractional Differential Problem

We give some background of fractional Cauchy problems and correct an existing result of Cauchy problem for the fractional differential inequality, and we also give an example to illustrate our statement.

Abstract degenerate fractional differential inclusions in Banach spaces

In the paper under review, we investigate a class of abstract degenerate fractional differential inclusions with Caputo derivatives. We consider subordinated fractional resolvent families generated by multivalued linear operators, which do have removable singularities at the origin. Semi-linear degenerate fractional Cauchy problems are also considered in this context.

Existence of Mild Solutions to a Cauchy Problem Presented by Fractional Evolution Equation with an Integral Initial Condition

In this article, we apply two new fixed point theorems to investigate the existence of mild solutions for a nonlocal fractional Cauchy problem with an integral initial condition in Banach spaces.

Fractional optimal control problem for variational inequalities with control constraints

In this article, the fractional optimal control problem for variational inequalities is considered. The fractional time derivative is considered in Riemann–Liouville sense. The existence and uniqueness of solutions to a class of fractional differential variational inequalities in a Sobolev space is studied. An application to a fractional variational inequality in a bounded domain with Dirichlet and Neumann boundary conditions is given. Constraints on controls are imposed. Necessary and sufficient optimality conditions for the fractional Cauchy problems with the quadratic performance functional are derived. Specifically, the Euler–Lagrange equations first order optimality condition with an adjoint problem are presented.

Applications of inverse tempered stable subordinators

The inverse tempered stable subordinator is a stochastic process that models power law waiting times between particle movements, with an exponential tempering that allows all moments to exist. This paper shows that the probability density function of an inverse tempered stable subordinator solves a tempered time-fractional diffusion equation, and its “folded” density solves a tempered time-fractional telegraph equation. Two explicit formulae for the density function are developed, and applied to compute explicit solutions to tempered fractional Cauchy problems, where a tempered fractional derivative replaces the first derivative in time. Several examples are given, including tempered fractional diffusion equations on bounded or unbounded domains, and the probability distribution of a tempered fractional Poisson process. It is shown that solutions to the tempered fractional diffusion equation have a cusp at the origin.

Fischer Decomposition and Cauchy–Kovalevskaya Extension in Fractional Clifford Analysis: The Riemann–Liouville Case

AbstractIn this paper we present the basic tools of a fractional function theory in higher dimensions by means of a fractional correspondence to the Weyl relations via fractional Riemann–Liouville derivatives. A Fischer decomposition, Almansi decomposition, fractional Euler and Gamma operators, monogenic projection, and basic fractional homogeneous powers are constructed. Moreover, we establish the fractional Cauchy–Kovalevskaya extension (FCK extension) theorem for fractional monogenic functions defined on ℝd. Based on this extension principle, fractional Fueter polynomials, forming a basis of the space of fractional spherical monogenics, i.e. fractional homogeneous polynomials, are introduced. We study the connection between the FCK extension of functions of the form xPl and the classical Gegenbauer polynomials. Finally, we present an example of an FCK extension.

On the existence of the mild solution for semilinear nonlocal fractional Cauchy problem

On the existence of the mild solution for semilinear nonlocal fractional Cauchy problem

Fractional Cauchy problems on compact manifolds

ABSTRACTWe investigate anomalous diffusion on compact Riemannian manifolds, modeled by time-changed Brownian motions. These stochastic processes are governed by equations involving the Laplace–Beltrami operator and a time-fractional derivative of order β ∈ (0, 1). We also consider time dependent random fields that can be viewed as random fields on randomly varying manifolds.

Existence of Mild Solutions to Nonlocal Fractional Cauchy Problems via Compactness

We obtain characterizations of compactness for resolvent families of operators and as applications we study the existence of mild solutions to nonlocal Cauchy problems for fractional derivatives in Banach spaces. We discuss here simultaneously the Caputo and Riemann-Liouville fractional derivatives in the cases0<α<1and1<α<2.

Abstract degenerate fractional differential inclusions in Banach spaces

In the paper under review, we investigate a class of abstract degenerate fractional differential inclusions with Caputo derivatives. We consider subordinated fractional resolvent families generated by multivalued linear operators, which do have removable singularities at the origin. Semi-linear degenerate fractional Cauchy problems are also considered in this context.

A fractional counting process and its connection with the Poisson process

We consider a fractional counting process with jumps of amplitude 1,2,...,k, withk∈N, whoseprobabilitiessatisfy a suitablesystemoffractionaldifference-differential equations. We obtain the moment generating function and the probability law of the result- ing process in terms of generalized Mittag-Leffler functions. We also discuss two equiv- alent representations both in terms of a compound fractional Poisson process and of a subordinator governed by a suitable fractional Cauchy problem. The first occurrence time of a jump of fixed amplitude is proved to have the same distribution as the waiting time of the first event of a classical fractional Poisson process, this extending a well-known property of the Poisson process. When k = 2 we also express the distribution of the first passage time of the fractional counting process in an integral form. Finally, we show that the ratios given by the powers of the fractional Poisson process and of the countingprocess over their means tend to 1 in probability.

On the solutions to a generalized fractional Cauchy problem

On the solutions to a generalized fractional Cauchy problem

Regularity of mild Solutions for fractional abstract Cauchy problem with order $${\alpha \in (1, 2)}$$ α ∈ ( 1 , 2 )

In this paper, we deal with the regularity of mild Solutions for fractional abstract Cauchy problem with order \({\alpha \in (1, 2)}\). Based on properties of solution operator and analytic solution operator, we obtain the sufficient condition under which a mild solution becomes a classical solution, and if the Cauchy problem has an analytic solution operator, we obtain more stronger results.

Semilinear fractional differential equations: global solutions, critical nonlinearities and comparison results

In this work we study several questions concerning to abstract fractional Cauchy problems of order $\alpha\in(0,1)$. Concretely, we analyze the existence of local mild solutions for the problem, and its possible continuation to a maximal interval of existence. The case of critical nonlinearities and corresponding regular mild solutions is also studied. Finally, by establishing some general comparison results, we apply them to conclude the global well-posedness of a fractional partial differential equation coming from heat conduction theory.

Sharp extensions and algebraic properties for solution families of vector-valued differential equations

In this paper we show the unexpected property that extension from local to global without loss of regularity holds for the solutions of a wide class of vector-valued differential equations, in particular for the class of fractional abstract Cauchy problems in the subdiffusive case. The main technique is the use of the algebraic structure of these solutions, which are defined by new versions of functional equations defining solution families of bounded operators. The convolution product and the double Laplace transform for functions of two variables are useful tools which we apply also to extend these solutions. Finally we illustrate our results with different concrete examples.

Reflected spectrally negative stable processes and their governing equations

This paper explicitly computes the transition densities of a spectrally negative stable process with index greater than one, reflected at its infimum. First we derive the forward equation using the theory of sun-dual semigroups. The resulting forward equation is a boundary value problem on the positive half-line that involves a negative Riemann-Liouville fractional derivative in space, and a fractional reflecting boundary condition at the origin. Then we apply numerical methods to explicitly compute the transition density of this space-inhomogeneous Markov process, for any starting point, to any desired degree of accuracy. Finally, we discuss an application to fractional Cauchy problems, which involve a positive Caputo fractional derivative in time.

An operator theoretical approach to Riemann-Liouville fractional Cauchy problem

This paper is concerned with developing an operator theory for Riemann-Liouville fractional Cauchy problem. Two notions, named Riemann-Liouville fractional resolvent and solution operator, are developed. Some of their properties are deduced. Moreover, the obtained results are applied to Riemann-Liouville fractional Cauchy problem.

Approximation of Semilinear Fractional Cauchy Problem

Abstract The semidiscretization methods for solving the Cauchy problem ( 𝐃 t α u ) ( t ) = A u ( t ) + J 1 - α f t , u ( t ) , t ∈ [ 0 , T ] , 0 < α < 1 , u ( 0 ) = u 0 , $(\mathbf {D}_{t}^{\alpha }u)(t) = A u(t) + J^{1-\alpha } f\big (t,u(t)\big ), \quad t \in [0,T], 0 < \alpha <1,\qquad u(0) = u^0,$ with operator A, which generates an analytic and compact resolution family { S α ( t , A ) } t ≥ 0 ${\lbrace S_{\alpha }(t,A)\rbrace _{t\ge 0}}$ , in a Banach space E are presented. It is proved that the compact convergence of resolvents implies the convergence of semidiscrete approximations to an exact solution. We give an analysis of a general approximation scheme, which includes finite differences and projective methods.

A Subordination Principle on Wright Functions and Regularized Resolvent Families

We obtain a vector-valued subordination principle forgα,gβ-regularized resolvent families which unified and improves various previous results in the literature. As a consequence, we establish new relations between solutions of different fractional Cauchy problems. To do that, we consider scaled Wright functions which are related to Mittag-Leffler functions, the fractional calculus, and stable Lévy processes. We study some interesting properties of these functions such as subordination (in the sense of Bochner), convolution properties, and their Laplace transforms. Finally we present some examples where we apply these results.