16,078 research outputs found

    Well-posedness and stability for fuzzy fractional differential equations

    Get PDF
    In this article, we consider the existence and uniqueness of solutions for a class of initial value problems of fuzzy Caputo–Katugampola fractional differential equations and the stability of the corresponding fuzzy fractional differential equations. The discussions are based on the hyperbolic function, the Banach fixed point theorem and an inequality property. Two examples are given to illustrate the feasibility of our theoretical results

    Impulsive boundary value problems for nonlinear implicit Caputo-exponential type fractional differential equations

    Get PDF
    This paper deals with existence and uniqueness of solutions to a class of impulsive boundary value problem for nonlinear implicit fractional differential equations involving the Caputo-exponential fractional derivative. The existence results are based on Schaefer’s fixed point theorem and the uniqueness result is established via Banach’s contraction principle. Two examples are given to illustrate the main results

    Three-point boundary value problems with delta Riemann-Liouville fractional derivative on time scales

    Get PDF
    In this paper, we establish the criteria for the existence and uniqueness of solutions of a three-point boundary value problem for a class of fractional differential equations on time scales. By using some well known fixed point theorems, sufficient conditions for the existence of solutions are established. An illustrative example is also presented

    Impulsive boundary value problems for nonlinear implicit Caputo-exponential type fractional differential equations

    Get PDF
    This paper deals with existence and uniqueness of solutions to a class of impulsive boundary value problem for nonlinear implicit fractional differential equations involving the Caputo-exponential fractional derivative. The existence results are based on Schaefer's fixed point theorem and the uniqueness result is established via Banach's contraction principle. Two examples are given to illustrate the main results

    Integral Inequalities and Differential Equations via Fractional Calculus

    Get PDF
    In this chapter, fractional calculus is used to develop some results on integral inequalities and differential equations. We develop some results related to the Hermite-Hadamard inequality. Then, we establish other integral results related to the Minkowski inequality. We continue to present our results by establishing new classes of fractional integral inequalities using a family of positive functions; these classes of inequalities can be considered as generalizations of order n for some other classical/fractional integral results published recently. As applications on inequalities, we generate new lower bounds estimating the fractional expectations and variances for the beta random variable. Some classical covariance identities, which correspond to the classical case, are generalised for any α ≥ 1 , β ≥ 1 . For the part of differential equations, we present a contribution that allow us to develop a class of fractional chaotic electrical circuit. We prove recent results for the existence and uniqueness of solutions for a class of Langevin-type equation. Then, by establishing some sufficient conditions, another result for the existence of at least one solution is also discussed

    Uniqueness, existence and regularity of solutions of integro-PDE in domains of R^n

    Get PDF
    The main goal of the thesis is to study integro-differential equations. Integro-differential equations arise naturally in the study of stochastic processes with jumps. These types of processes are of particular interest in finance, physics and ecology. In the first part of my thesis, we study interior regularity for the regional fractional Laplacian operator. We first obtain the integer order differentiability of the regional fractional Laplacian. We further extend the integer order differentiability to the fractional order of the regional fractional Laplacian. Schauder estimates for the regional fractional Laplacian are also provided. In the second and third parts of my thesis, we consider uniqueness and existence of viscosity solutions for a class of nonlocal equations. This class of equations includes Bellman-Isaacs equations containing operators of L\'evy type with measures depending on xx and control parameters, as well as elliptic nonlocal equations that are not strictly monotone in the uu variable. In the fourth part of my thesis, we obtain semiconcavity of viscosity solutions for a class of degenerate elliptic integro-differential equations in Rn\mathbb R^n. This class of equations includes Bellman equations containing operators of L\'evy-It\^o type. H\"{o}lder and Lipschitz continuity of viscosity solutions for a more general class of degenerate elliptic integro-differential equations are also proved. In the last part of my thesis, we study interior regularity of viscosity solutions of non-translation invariant nonlocal fully nonlinear equations with Dini continuous terms. We obtain CσC^{\sigma} regularity estimates for the nonlocal equations by perturbative methods and a version of a recursive Evans-Krylov theorem.Ph.D

    Inverse source problems for positive operators. I: Hypoelliptic diffusion and subdiffusion equations

    Get PDF
    A class of inverse problems for restoring the right-hand side of a parabolic equation for a large class of positive operators with discrete spectrum is considered. The results on existence and uniqueness of solutions of these problems as well as on the fractional time diffusion (subdiffusion) equations are presented. Consequently, the obtained results are applied for the similar inverse problems for a large class of subelliptic diffusion and subdiffusion equations (with continuous spectrum). Such problems are modelled by using general homogeneous left-invariant hypoelliptic operators on general graded Lie groups. A list of examples is discussed, including Sturm-Liouville problems, differential models with involution, fractional Sturm-Liouville operators, harmonic and anharmonic oscillators, Landau Hamiltonians, fractional Laplacians, and harmonic and anharmonic operators on the Heisenberg group. The rod cooling problem for the diffusion with involution is modelled numerically, showing how to find a "cooling function", and how the involution normally slows down the cooling speed of the rod.Comment: 26 pages, 7 figures. arXiv admin note: text overlap with arXiv:1812.0133
    • …
    corecore