Search

In this manuscript, our work was about a qualitative study for a class of multi-complex orders nonlinear fractional differential equations (FDEs). Our methodology utilized the topological degree theory and studied a novel operator tailored for non-singular FDEs with T-Riemann-Liouville (T-RL) fractional order derivatives. The primary objective was to prove the existence and uniqueness of solutions for both initial and boundary value problems within the intricated framework. With the help of topological degree theory, we contributed to a wider understanding of the structural aspects governing the behavior of the considered FDE. The novel operator proposing for non-singular FDEs added a unique dimension to our analytical problem, offering a versatile and effective means of addressing the challenges posed by these complex systems for their theoretical analysis. For the practical implications of our theoretical framework, we presented two concrete examples that reinforced and elucidated the key concepts introduced. These examples underscored our approach's viability and highlighted its potential applications in diverse scientific and engineering domains. Through this comprehensive exploration, we aimed to contribute significantly to advancing the theoretical foundation related to the study of multi-complex nonlinear FDEs. Moreover, a fixedtime terminal sliding mode control (TSMC) has been developed. This proposed control strategy for eliminating leukemic cells while maintaining normal cells was based on a chemotherapeutic treatment that was not only effective but also widely acknowledged to be safe. This strategy was evaluated using the fixed-time Lyapunov stability theory, and simulations were included to illustrate its performance in terms of tracking and convergence. [ABSTRACT FROM AUTHOR]

Addressing the boundary value problems of fractional-order differential equations hold significant importance due to their applications in various fields. The aim of this paper was to approximate solutions for a class of boundary value problems involving Caputo fractional-order differential equations employing the AA-iterative scheme. Moreover, the stability and data dependence results of the iterative scheme were given for a certain class of mappings. Finally, a numerical experiment was illustrated to support the results presented herein. The results presented in this paper extend and unify some well-known comparable results in the existing literature. [ABSTRACT FROM AUTHOR]

We investigate the criteria for both the existence and uniqueness of solutions within a nonlinear coupled system of Hilfer-Hadamard sequential fractional differential equations featuring varying orders. This system is complemented by nonlocal coupled Hadamard fractional integral boundary conditions. The desired outcomes are attained through the application of well-established fixed-point theorems. It is underscored that the fixed-point approach serves as an effective method for establishing both the existence and uniqueness of solutions to boundary value problems. The results obtained are further demonstrated and validated through illustrative examples. [ABSTRACT FROM AUTHOR]

This paper discusses the existence of solutions for fractional differential equations with nonlocal boundary conditions (NFDEs) under essential assumptions. The boundary conditions incorporate a point 0 ≤ c < d and fixed points at the end of the interval [0, d]. For i = 0, 1, the boundary conditions are as follows: ai, bi > 0, a0p(c) = -b0p(d), a1p'(c) = -b1p'(d). Furthermore, the research aims to expand the usability and comprehension of these results to encompass not just NFDEs but also classical fractional differential equations (FDEs) by using the Krasnoselskii fixedpoint theorem and the contraction principle to improve the completeness and usefulness of the results in a wider context of fractional differential equations. We offer examples to demonstrate the results we have achieved. [ABSTRACT FROM AUTHOR]

In this work, we study a posteriori error analysis of a general class of fractional initial value problems and fractional boundary value problems. A Petrov-Galerkin spectral method is adopted as the discretization technique in which the generalized Jacobi functions are utilized as basis functions for constructing efficient spectral approximations. The unique solvability of the weak problems is established by verifying the Babuška-Brezzi inf-sup condition. Then, we introduce some residual-type a posteriori error estimators, and deduce their efficiency and reliability in properly weighted Sobolev space. Numerical examples are given to illustrate the performance of the obtained error estimators. [ABSTRACT FROM AUTHOR]

In this work, we investigate the existence, uniqueness and the stability of solutions to the boundary value problem (BVP) of Caputo fractional differential equations of variable order by converting it into an equivalent standard Caputo BVP of the fractional constant order with the help of the generalized intervals and piecewise constant functions. The results obtained in this interesting study are novel and worthy based on the Krasnoselskii fixed point theorem and the Banach contraction principle. The Ulam-Hyers stability of the given variable-order Caputo fractional boundary value problem is established. A numerical examples is given at the end to support and validate the potentiality of our obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and Ulam stability results for a class of boundary value problems for implicit Riesz-Caputo fractional differential equations with non-instantaneous impulses involving both retarded and advanced arguments. The result are based on Monch fixed point theorem associated with the technique of measure of noncompactness. An illustrative example is given to validate our main results. [ABSTRACT FROM AUTHOR]

In this work, we consider the fractional Stefan-type problem in a Lipschitz bounded domain Ω ⊂ ℝd with time-dependent Dirichlet boundary condition for the temperature ϑ = ϑ(x, t), ϑ = g on Ωc×]0, T[, and initial condition η0 for the enthalpy η = η(x, t), given in Ω×]0, T[ by ∂η/∂t + LsAϑ = f with η ∈ β(ϑ), where LsA is an anisotropic fractional operator defined in the distributional sense by sAu, v> = ∫ℝd ADsu · Ds v dx, β is a maximal monotone graph, A(x) is a symmetric, strictly elliptic and uniformly bounded matrix, and Ds is the distributional Riesz fractional gradient for 0 < s < 1. We show the existence of a unique weak solution with its corresponding weak regularity. We also consider the convergence as s ↗ 1 towards the classical local problem, the asymptotic behaviour as t → ∞, and the convergence of the two-phase Stefan-type problem to the one-phase Stefan-type problem by varying the maximal monotone graph β. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of solutions to a nonlinear coupled systems of (k;φ')-Hilfer fractional differential equations supplemented with nonlocal integromulti- point boundary conditions. We make use of the Banach contraction mapping principle to obtain the uniqueness result, while the existence results are proved with the aid of Krasnosel'skii's fixed point theorem and Leray-Schauder alternative for the given problem. Examples demonstrating the application of the abstract results are also presented. Our results are of quite general nature and specialize in several new results for appropriate values of the parameters β1; β2; and the function φ involved in the problem at hand. [ABSTRACT FROM AUTHOR]

In this paper, we formulate and prove the weak and strong maximum principles for a general parabolic-type fractional differential operator with the Riemann-Liouville time-fractional derivative of distributed order. The proofs of the maximum principles are based on an estimate of the Riemann-Liouville fractional derivative at its maximum point that was recently derived by the authors. Some a priori norm estimates for solutions to initial-boundary value problems for linear and nonlinear fractional diffusion equations of distributed order and uniqueness results for these problems are presented. [ABSTRACT FROM AUTHOR]

In this work, the existence of solutions for nonlinear hybrid fractional integro-differential equations involving generalized proportional fractional (GPF) derivative of Caputo-Liouville-type and multi-term of GPF integrals of Reimann-Liouville type with Dirichlet boundary conditions is investigated. The analysis is accomplished with the aid of the Dhage's fixed point theorem with three operators and the lower regularized incomplete gamma function. Further, the uniqueness of solutions and their Ulam-Hyers-Rassias stability to a special case of the suggested hybrid problem are discussed. For the sake of corroborating the obtained results, an illustrative example is presented. [ABSTRACT FROM AUTHOR]

We investigate the initial-boundary value problems for a fourth-order differential equation within the powerful fractional Dzherbashian-Nersesian operator (FDNO). Boundary conditions considered in this manuscript are of the Samarskii-Ionkin type. The solutions obtained here are based on a series expansion using Riesz basis in a space corresponding to a non-self-adjoint spectral problem. Conditional to some regularity, consistency, alongside orthogonality dependence, the existence and uniqueness of the obtained solutions are exhibited by using Fourier method. Acquired results here are more general than those obtained by making use of conventional fractional operators such as fractional Riemann-Liouville derivative (FRLD), fractional Caputo derivative (FCD) and fractional Hilfer derivative (FHD). [ABSTRACT FROM AUTHOR]

The existence-uniqueness criteria of nonlinear fractional integro-differential equations of variable order with multiterm boundary value conditions are considered in this work. By utilizing the concepts of generalized intervals combined with the piecewise constant functions, we transform our problem into usual Caputo's fractional differential equations of constant order. We develop the necessary criteria for assuring the solution's existence and uniqueness by applying Schauder and Banach fixed point theorem. We also examine the stability of the derived solution in the Ulam-Hyers-Rassias (UHR) sense and provide an example to demonstrate the credibility of the results. [ABSTRACT FROM AUTHOR]

This paper studies a class of fourth point singular boundary value problem of p-Laplacian operator in the setting of a specific kind of conformable derivatives. By using the upper and lower solutions method and fixed point theorems on cones., necessary and sufficient conditions for the existence of positive solutions are obtained. In addition, we investigate the dependence of the solution on the order of the conformable differential equation and on the initial conditions. [ABSTRACT FROM AUTHOR]

In this study, the two-point boundary value problem is considered for the variable fractional order differential equation with causal operator. Under the definition of the Caputo-type variable fractional order operators, the necessary inequality and the existence results of the solution are obtained for the variable order fractional linear differential equations according to Arzela–Ascoli theorem. Then, based on the proposed existence results and the monotone iterative technique, the existence of the extremal solution is studied, and the relative results are obtained based on the lower and upper solution. Finally, an example is provided to illustrate the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

We study a fractional diffusion problem in the divergence form in one space dimension. We define a notion of the viscosity solution. We prove existence of viscosity solutions to the fractional diffusion problem with the Dirichlet boundary values by Perron's method. Their uniqueness follows from a proper maximum principle. We also show a stability result and basic regularity of solutions. [ABSTRACT FROM AUTHOR]

We study the inverse boundary value problem for fractional diffusion in a multilayer composite medium. Given data in the right boundary of the second layer, the problem is to recover the temperature distribution in the first layer, which is inaccessible for measurement. The problem is ill-posed and we propose a Fourier spectral approach to achieve Hölder approximations. The convergence analysis is performed in both the $L^{2}$ - and $L^{\infty }$ -settings. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence of solutions to a boundary value problem for differential equations of variable order. Our results are based on the Schauder fixed point theorem. Some examples are given to illustrate the effectiveness of our results. [ABSTRACT FROM AUTHOR]

We study the existence of positive solutions for a system of nonlinear Riemann- Liouville fractional differential equations subject to multi-point boundary conditions which contain fractional derivatives, where the nonlinearities are nonnegative and singular functions. In the proof of our main results we use the Guo-Krasnosel'skii fixed point theorem. [ABSTRACT FROM AUTHOR]

In this work, we propose novel discretizations of the spectral fractional Laplacian on bounded domains based on the integral formulation of the operator via the heat-semigroup formalism. Specifically, we combine suitable quadrature formulas of the integral with a finite element method for the approximation of the solution of the corresponding heat equation. We derive two families of discretizations with order of convergence depending on the regularity of the domain and the function on which the spectral fractional Laplacian is acting. Our method does not require the computation of the eigenpairs of the Laplacian on the considered domain, can be implemented on possibly irregular bounded domains, and can naturally handle different types of boundary constraints. Various numerical simulations are provided to illustrate performance of the proposed method and support our theoretical results. [ABSTRACT FROM AUTHOR]

To characterize the Brownian motion in a bounded domain , it is well known that the boundary conditions of the classical difuusion equation just rely on the given information of the solution along the boundary of a domain; in contrast, for the Lévy ights or tempered Lévy ights in a bounded domain, the boundary conditions involve the information of a solution in the complementary set of , i.e., Rnn , with the potential reason that paths of the corresponding stochastic process are discontinuous. Guided by probability intuitions and the stochastic perspectives of anomalous difusion, we show the reasonable ways, ensuring the clear physical meaning and well-posedness of the partial differential equations (PDEs), of specifying \boundary" conditions for space fractional PDEs modeling the anomalous difusion. Some properties of the operators are discussed, and the well-posednesses of the PDEs with generalized boundary conditions are proved. [ABSTRACT FROM AUTHOR]

We study the problem ... where s ∈ (0; 1) is a fixed parameter, (-Δ)s is the fractional Laplacian in RN, q > p ≥ N+2s/N-2s and N > 2s. For every s &#8712: (0; 1), we establish regularity results of solutions of above equation (whenever solution exists) and we show that every solution is a classical solution. Next, we derive certain decay estimate of solutions and the gradient of solutions at infinity for all s ∈ (0; 1). Using those decay estimates, we prove Pohozaev type identity in RN and we show that the above problem does not have any solution when p = N+2s/ N-2s . We also discuss radial symmetry and decreasing property of the solution and prove that when p > N+2s/N-2s, the above problem admits a solution . Moreover, if we consider the above equation in a bounded domain with Dirichlet boundary condition, we prove that it admits a solution for every p > N+2s/N-2s and every solution is a classical solution. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following perturbed nonlocal elliptic equation {-LKu = ηu + f(x, u) + g(x, u), x ∋ Ω, u = 0; x ∋ RN\Ω, where Ω is a smooth bounded domain in RN, η is a real parameter and g is a non-odd perturbation term. If f is odd in u and satisfies various superlinear growth conditions at infinity in u, infinitely many solutions are obtained in spite of the lack of the symmetry of this problem for any η ∋ R. The results obtained in this paper may be seen as natural extensions of some classical theorems to the case of nonlocal operators. Moreover, the methods used in this paper can be also applied to obtain some new results for the classical Laplace equation with Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we consider an initial boundary value problem for nonlocal-in-time parabolic equations involving a nonlocal in time derivative. We first show the uniqueness and existence of the weak solution of the nonlocal-in-time parabolic equation, and also the spatial smoothing properties. More-over, we develop a new framework to study the local limit of the nonlocal model as the horizon parameter δ approaches 0. Exploiting the spatial smoothing properties, we develop a semi-discrete scheme using standard Galerkin finite element method for the spatial discretization, and derive error estimates dependent on data smoothness. Finally, extensive numerical results are presented to illustrate our theoretical findings. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain the existence of solution for the fractional p-Laplacian Dirichlet problem with mixed derivatives t D Tα(|0 D tα u( t)| p-20 D tα u( t)) = f( t, u( t)), t ∈ [0, T], u(0) = u( T) = 0, where 1/ p < α < 1, 1 < p < ∞ and f : [0, T] × ℝ → ℝ is a Carathéodory function which satisfies some growth conditions. We obtain the existence of nontrivial solutions by using the direct method in variational methods and mountain pass theorem. [ABSTRACT FROM AUTHOR]

We investigate a class of semilinear parabolic and elliptic problems with fractional dynamic boundary conditions. We introduce two new operators, the so-called fractional Wentzell Laplacian and the fractional Steklov operator, which become essential in our study of these nonlinear problems. Besides giving a complete characterization of well-posedness and regularity of bounded solutions, we also establish the existence of finite-dimensional global attractors and also derive basic conditions for blow-up. [ABSTRACT FROM AUTHOR]

A distributed order fractional diffusion equation is considered. Distributed order derivatives are fractional derivatives that have been integrated over the order of the derivative within a given range. In this paper, sub-diffusive cases are considered. That is, the order of the time derivative ranges from zero to one. The equation is solved for Dirichlet, Neumann and Cauchy boundary conditions. The time dependence for each of the three cases is found to be a functional of the diffusion parameter. This functional is shown to have decay properties. Upper and lower bounds are computed for the functional. Examples are also worked out for comparative decay rates. [ABSTRACT FROM AUTHOR]