Hilfer Derivative Research Articles

A study on existence and stability analysis of implicit k,Ψ$$ \left(k,\Psi \right) $$‐Hilfer fractional differential equations and application to RC$$ \mathcal{RC} $$ circuit model

In the present study, we establish the existence and uniqueness of solutions for nonlinear initial value problems and nonlocal boundary value problems associated with implicit fractional differential equations involving ‐Hilfer derivative operator. Furthermore, we explore the stability properties of these solutions in the sense of the Ulam–Hyers, Ulam–Hyers–Rassias, and their generalized stability concepts by proving and applying generalized Gronwall inequality. Lastly, we present the fractional electric circuit model as an example to show the practical applicability of our main results.

On a class of generalized capillarity system involving fractional ψ$$ \psi $$‐Hilfer derivative with p(·)$$ p\left(\cdotp \right) $$‐Laplacian operator

This research delves into a comprehensive investigation of a class of ‐Hilfer generalized fractional nonlinear differential system originated from a capillarity phenomena with Dirichlet boundary conditions, focusing on issues of existence and multiplicity of nonnegative solutions. The nonlinearity of the problem, in general, does not satisfy the Ambrosetti–Rabinowitz type condition. We use minimization arguments of Nehari manifold together with variational approach to show the existence and multiplicity of positive solutions of our problem with respect to the parameter in appropriate fractional ‐Hilfer spaces. Our main result is novel, and its investigation will enhance the scope of the literature on coupled systems of fractional ‐Hilfer generalized capillary phenomena.

On Leibniz type rule for generalized fractional derivatives

In this paper, we present a new Leibniz-type rule for two variants of the most generalized fractional derivatives; the regularized ψ-Hilfer derivative and the (k,ψ)-Hilfer derivative. As a special case, we recover the Leibniz-type rule associated to some well known fractional derivatives.

Existence results for a coupled system of fractional stochastic differential equations involving Hilfer derivative

Abstract In this paper, we consider a coupled system of fractional stochastic differential equations involving the Hilfer derivative of order 1 2 < α < 1 {\frac{1}{2}<\alpha<1} . Under some assumptions, we prove the existence of mild solutions for our system based on Perov’s and Schaefer’s fixed point theorems. An example illustrating our result is provided.

An Approach to Solve Gas Dynamic Equation by Fuzzy Hilfer Fractional Differential Equation

In this work, Adomian decomposition method (ADM) based on Hilfer derivative is proposed to solve nonlinear fuzzy fractional gas dynamic equation. We reform the provided fractional gas dynamic equation into fuzzy Hilfer fractional differential equation. Subsequently with description of Hilfer fractional derivative and suitable initial conditions under fuzzy sense, we obtain approximate solution to the given problem. This method provides an analytical solution in the form of infinite power series. The behavior of the solution is illustrated using graphical representation.

A Class of Quasilinear Equations with Hilfer Derivatives

A Class of Quasilinear Equations with Hilfer Derivatives

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Hilfer fractional derivative is an important and interesting operator in fractional calculus, and it can be applicable in pure theories and other fields. It yields to other notable definitions, Ψ-Hilfer, (k,Ψ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(k,\\Psi )$\\end{document}-Hilfer derivatives, etc. Motivated by the concepts of the proportional fractional derivative and (k,Ψ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(k,\\Psi )$\\end{document}-Hilfer fractional derivative, we first introduce new definitions of integral and derivative, termed the (ρ,k,Ψ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\rho ,k,\\Psi )$\\end{document}-proportional integral and (ρ,k,Ψ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\rho ,k,\\Psi )$\\end{document}-proportional Hilfer fractional derivative. This type of fractional derivative is advantageous as it aligns with earlier studies on fractional differential equations. Additionally, we present a more generalized version of the (ρ,α,β,k,r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\rho ,\\alpha ,\\beta ,k,r)$\\end{document}-resolvent family, followed by an exploration of its properties. By analyzing the generalized resolvent family, we examine the existence of mild solutions to the (ρ,k,Ψ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\rho ,k,\\Psi )$\\end{document}-proportional Hilfer fractional Cauchy problem, supported by an illustrative example to show the main result.

Study on a Nonlocal Fractional Coupled System Involving (k,ψ)-Hilfer Derivatives and (k,ψ)-Riemann–Liouville Integral Operators

This paper deals with a nonlocal fractional coupled system of (k,ψ)-Hilfer fractional differential equations, which involve, in boundary conditions, (k,ψ)-Hilfer fractional derivatives and (k,ψ)-Riemann–Liouville fractional integrals. The existence and uniqueness of solutions are established for the considered coupled system by using standard tools from fixed point theory. More precisely, Banach and Krasnosel’skiĭ’s fixed-point theorems are used, along with Leray–Schauder alternative. The obtained results are illustrated by constructed numerical examples.

Controllability of psi-Hilfer fractional differential equations with infinite delay via measure of noncompactness

In this article, we study the controllability of ψ-Hilfer fractional differential equations with infinite delay. Sufficient conditions for controllability results are obtained by using the notion of the measure of noncompactness and the Mönch fixed point theorem. The novel feature of this study is to inquire into the controllability notion by using ψ-Hilfer fractional derivative, the generalized variant of the Hilfer derivative. Finally, we provide a numerical example to illustrate our main result.

On Nonlinear Analysis for Multi-term Delay Fractional Differential Equations Under Hilfer Derivative

AbstractIn this manuscript, a class of multi-term delay fractional differential equations (FDEs) under the Hilfer derivative is considered. Some newly updated results are established under boundary conditions. For the required results, we utilize the fixed point theory and tools of the nonlinear functional analysis. Further keeping in mind the importance of stability results, we develop some adequate results about the said aspect. The Hyers-Ulam (H-U)-type concept is used to derive the required stability for the solution of the considered problem. Finally, by appropriate test problems, we justify our findings.

Separable solutions of the Black-Scholes equation with three different time fractional-order derivatives

In this work, we use the invariant subspace method to obtain separable solutions for the well-known time-fractional Black-Scholes equation (tF-BSE) under three different kinds of fractional derivatives that are (i) Caputo derivative, (ii) regularized Prabhakar derivative, and (iii) Hilfer derivative. Also, the comparison of the obtained solutions of tF-BSE with those mentioned above three different fractional-order derivatives has been discussed. In addition, this work shows that generalized separable exact solutions include the two- and three-parameters of the Mittag-Leffler, the Euler-gamma, exponential, and polynomial functions. Additionally, we compare the obtained separable solutions in two-dimensional graphically under considered fractional-order derivatives for various values of fractional-order α, α ϵ (0, 1].

Analytic Resolving Families of Operators for Linear Equations with Hilfer Derivative

Analytic Resolving Families of Operators for Linear Equations with Hilfer Derivative

Qualitative Analysis of RLC Circuit Described by Hilfer Derivative with Numerical Treatment Using the Lagrange Polynomial Method

This paper delves into an examination of the existence, uniqueness, and stability properties of a non-local integro-differential equation featuring the Hilfer fractional derivative with order ω∈(1,2) for the RLC model. Based on Schaefer’s fixed point theorem and Banach’s contraction principle, the existence and uniqueness results are established. Furthermore, Ulam–Hyers and Ulam–Hyers–Rassias stability results for the boundary value problem of the RLC model are discussed. To showcase the practicality and efficacy of our theoretical findings, a two-step Lagrange polynomial interpolation method is applied to solve some numerical examples.

Inverse coefficient problems for a time-fractional wave equation with the generalized Riemann-Liouville time derivative

This paper considers the inverse problem of determining the time-dependent coeffiicient in the fractional wave equation with Hilfer derivative. In this case, the direct problem is initial-boundary value problem for this equation with Cauchy type initial and nonlocal boundary conditions. As overdetermination condition nonlocal integral condition with respect to direct problem solution is given. By the Fourier method, this problem is reduced to equivalent integral equations. Then, using the Mittag-Leffler function and the generalized singular Gronwall inequality, we get apriori estimate for solution via unknown coefficient which we will need to study of the inverse problem. The inverse problem is reduced to the equivalent integral of equation of Volterra type. The principle of contracted mapping is used to solve this equation. Local existence and global uniqueness results are proved.

Mild solutions for impulsive fractional differential inclusions with Hilfer derivative in Banach spaces

Mild solutions for impulsive fractional differential inclusions with Hilfer derivative in Banach spaces

Inverse coefficient problem for the time‐fractional diffusion equation with Hilfer operator

In this paper, the inverse problem of determining the time‐dependent reaction–diffusion coefficient in the Cauchy problem for the time‐fractional diffusion equation with the Hilfer operator by a single observation at the point of the diffusion process is studied. To represent the solution of the direct problem, the fundamental solution of this equation is constructed and properties of this solution are investigated. The fundamental solution contains Fox's ‐functions widely used in fractional calculus. The fundamental solution for the fractional diffusion operator with the Hilfer derivative is also obtained in terms of the Yang ‐function, and it is shown that they coincide. Using estimates of the fundamental solution and its derivatives, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown coefficient, which will be used in investigation of inverse problem. The inverse problem is reduced to the equivalent integral equation. For solving this equation, the contracted mapping principle is applied. The local existence and global uniqueness results and the conditional stability estimate of solution are proven.

Maximum Principle for Nonlinear Fractional Differential Equations with the Hilfer Derivative

In this paper, two significant inequalities for the Hilfer fractional derivative of a function in the space ACγ([0,b],Rn), 0≤γ≤1 are obtained. We first verified the extremum principle for the Hilfer fractional derivative. In addition, we estimated the Hilfer derivative of a function at its extreme points. Furthermore, we derived and proved a maximum principle for a nonlinear Hilfer fractional differential equation. Finally, we analyzed the solutions of a nonlinear Hilfer fractional differential equation. Our results generalize and extend some obtained theorems on this topic.

The Regional Enlarged Observability for Hilfer Fractional Differential Equations

In this paper, we investigate the concept of regional enlarged observability (ReEnOb) for fractional differential equations (FDEs) with the Hilfer derivative. To proceed this, we develop an approach based on the Hilbert uniqueness method (HUM). We mainly reconstruct the initial state ν01 on an internal subregion ω from the whole domain Ω with knowledge of the initial information of the system and some given measurements. This approach shows that it is possible to obtain the desired state between two profiles in some selective internal subregions. Our findings develop and generalize some known results. Finally, we give two examples to support our theoretical results.

Existence of Mild Solutions for Nonlocal Evolution Equations with the Hilfer Derivatives

The existence of mild solutions for Hilfer fractional evolution equations with nonlocal conditions in a Banach space is investigated in this manuscript. No assumptions about the compactness of a function or the Lipschitz continuity of a nonlinear function are imposed on the nonlocal item and the nonlinear function, respectively. However, we assumed that the nonlocal item is continuous, the nonlinear term is continuous and satisfies some specified assumptions, and the associated semigroup is compact. Our theorems are proved by means of approximate techniques, semigroup methods, and fixed point theorem. These methods are useful for fixing the noncompactness of operators caused by some specified given assumptions on this paper. The results obtained here improve some known results. Finlay, two examples are presented for illustration of our main results.

A boundary problem for the time-fractional Hallaire–Luikov moisture transfer equation with Hilfer derivative

A boundary problem for the time-fractional Hallaire–Luikov moisture transfer equation with Hilfer derivative