Fractional Differential Problem Research Articles

An interior penalty method for a parabolic complementarity problem involving a fractional Black-Scholes operator

In this paper, an interior penalty method is proposed to solve a parabolic complementarity problem involving fractional Black–Scholes operator arising in pricing American options under a geometric Lévy process. The complementarity problem is first reformulated as a fractional partial differential variational inequality problem using the representations of fractional order operators and appropriate mathematical techniques. A penalty equation is then proposed to approximate the variational inequality problem by introducing a novel interior-point based penalty term. The existence and uniqueness of the solution to the penalized problem are proved, and an upper bound on the distance between the solutions to the penalty equation and the variational inequality problem is established. To test our method, we discretize the penalty equation by a finite difference method in space and the Crank–Nicolson method in time. We then present numerical experimental results to demonstrate the usefulness and effectiveness for the interior penalty method.

On the Equivalence between Differential and Integral Forms of Caputo-Type Fractional Problems on Hölder Spaces

As claimed in many papers, the equivalence between the Caputo-type fractional differential problem and the corresponding integral forms may fail outside the spaces of absolutely continuous functions, even in Hölder spaces. To avoid such an equivalence problem, we define a “new” appropriate fractional integral operator, which is the right inverse of the Caputo derivative on some Hölder spaces of critical orders less than 1. A series of illustrative examples and counter-examples substantiate the necessity of our research. As an application, we use our method to discuss the BVP for the Langevin fractional differential equation dψβ,μdtβdψα,μdtα+λx(t)=f(t,x(t)),t∈[a,b],λ∈R, for f∈C[a,b]×R and some critical orders β,α∈(0,1), combined with appropriate initial or boundary conditions, and with general classes of ψ-tempered Hilfer problems with ψ-tempered fractional derivatives. The BVP for fractional differential problems of the Bagley–Torvik type was also studied.

Duality of fractional derivatives: On a hybrid and non-hybrid inclusion problem

Abstract The main goal of this paper is to investigate a newly proposed hybrid and hybrid inclusion problem consisting of fractional differential problems involving two different fractional derivatives of order μ, Caputo and Liouville–Riemann operators, with multi-order mixed Riemann–Liouville integro-derivative conditions. Although α is between one and two, we need three boundary value conditions to find the integral equation. The study investigates the results of existence for hybrid, hybrid inclusion, and non-hybrid inclusion problems by employing several analytical approaches, including Dhage’s technique, α - ψ {\alpha-\psi} -contractive mappings, fixed points, and endpoints of the product operators. To further illustrate our findings, we present three examples.

Solvability of a fractional differential equation multipoint boundary value problem at resonance in ℝn

Solvability of a fractional differential equation multipoint boundary value problem at resonance in ℝn

High order numerical method for a subdiffusion problem

We consider a subdiffusive fractional differential problem characterized by an equation that incorporates a time Riemann-Liouville fractional derivative of order 1−α, α∈(0,1), on its right-hand side, while the diffusive coefficient is allowed to vary with both space and time. An high order numerical method for the subdiffusion problem is derived based on the fractional splines of degree β∈(1,2]. The main purpose of this work is to apply fractional splines for approximating the fractional integral in the definition of the Riemann-Liouville fractional derivative, and hence explain how to solve the subdiffusion problem using this approach. It is discussed how the rate of convergence of the numerical method depends on the solution, the degree of the spline and the order of the fractional derivative.

Legendre-Galerkin spectral algorithm for fractional-order BVPs: Application to the Bagley-Torvik equation

<p>Herein, we provide an efficient spectral Galerkin algorithm, according to a special type of shifted Legendre basis for finding a semi-analytic solution to the Liouville-Caputo fractional boundary value problem. The algorithm’s main goal is to transform the fractional differential problem into a linear system with efficiently invertible, well-structured matrices. The convergence rates of the algorithm are carefully obtained as well as the error bound.</p>

Stability of a fractional heat equation with memory

Of concern is a fractional differential problem of order between zero and one. The model generalizes an existing well-known problem in heat conduction theory with memory. First, we justify the replacement of the first order derivative by a fractional one. Then, we establish a Mittag-Leffler stability result for a class of heat flux relaxation functions. We will combine the energy method with some properties from fractional calculus.

A Qualitative Analysis of a Non-Linear Coupled System under Two Types of Fractional Derivatives along with Mixed Boundary Conditions

This work addresses the qualitative analysis of a novel non-linear coupled system of fractional differential problems (FDPs) using Caputo and Liouville–Riemann fractional derivatives. Fractional calculus has demonstrated significant applicability across various fields, including financial systems, optimal control, epidemiological models, chaotic systems, and engineering. The proposed model builds on existing research by formulating a non-linear coupled fractional boundary value problem with mixed boundary conditions. The primary advantages of our method include its ability to capture the dynamics of complex systems more accurately and its flexibility in handling different types of fractional derivatives. The model’s solution was derived using advanced mathematical techniques, and the results confirmed the existence and uniqueness of the solutions. This approach not only generalizes classical differential equation methods but also offers a robust framework for modeling real-world phenomena governed by fractional dynamics. The study concludes with the validation of the theoretical findings through illustrative examples, highlighting the method’s efficacy and potential for further applications.

An efficient collocation technique based on operational matrix of fractional-order Lagrange polynomials for solving the space-time fractional-order partial differential equations

In this research, we propose a novel and fast computational technique for solving a class of space-time fractional-order linear and non-linear partial differential equations. Caputo-type fractional derivatives are considered. The proposed method is based on the operational and pseudo-operational matrices for the fractional-order Lagrange polynomials. To carry out the method, first, we find the integer and fractional-order operational and pseudo-operational matrix of integration. The collocation technique and obtained operational and pseudo-operational matrices are then used to generate a system of algebraic equations by reducing the given space-time fractional differential problem. The resultant algebraic system is then easily solved by Newton's iterative methods. The upper bound of the fractional-order operational matrix of integration is also provided, which confirms the convergence of fractional-order Lagrange polynomial's approximation. Finally, some numerical experiments are conducted to demonstrate the applicability and usefulness of the suggested numerical scheme.

On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems

Abstract The main goal of this article is to study the existence and uniqueness of periodic solutions for the implicit problem with nonlinear fractional differential equation involving the Caputo tempered fractional derivative. The proofs are based upon the coincidence degree theory of Mawhin. To show the efficiency of the stated result, two illustrative examples will be demonstrated.

The computational orthogonal shifted Legendre–Galerkin approach for handling fractional delay differential problems via adapting fractional M-derivative

This paper presents a numerical procedure for handling delay fractional differential problems where the derivative is defined using the M-fractional approach. The proposed scheme modus operandi is based on the shifted Legendre–Galerkin procedure, which is a powerful tool for solving complex differential models of generalized fractional derivatives. The method involves constructing a series of Legendre polynomials that form the basis functions for approximating the solution of the required problem. The coefficients of the series are obtained after solving an algebraic system of linear types that results from the application of the Galerkin practice. The numerical accuracy and convergence assessment are also presented together with various results. Simulations-based analyses are realized to validate the truthfulness and exactness of the process. The results manifest that the M-derivatives and the Galerkin practice provide alternative innovative approaches for handling M-delay fractional problems. Several keynotes and future recommendations are exhibited at the last with some selected references.

Numerical approximation of the space-time fractional diffusion problem

Fractional differential equations have become central tools for the accurate modeling of real-world phenomena in various fields. This work focuses on the discretization of the space-time fractional diffusion problem with Caputo derivative in time and Riesz-Caputo derivative in space. We introduce a collocation method based on a B-spline representation of the solution. This approach strategically exploits the symmetry properties of both the spline basis functions and the Riesz-Caputo operator, resulting in an efficient method for solving the given fractional differential problem. Preliminary numerical tests are presented to validate the proposed collocation method.

High-Order Chebyshev Pseudospectral Tempered Fractional Operational Matrices and Tempered Fractional Differential Problems

This paper focuses on presenting an accurate, stable, efficient, and fast pseudospectral method to solve tempered fractional differential equations (TFDEs) in both spatial and temporal dimensions. We employ the Chebyshev interpolating polynomial for g at Gauss–Lobatto (GL) points in the range [−1,1] and any identically shifted range. The proposed method carries with it a recast of the TFDE into integration formulas to take advantage of the adaptation of the integral operators, hence avoiding the ill-conditioning and reduction in the convergence rate of integer differential operators. Via various tempered fractional differential applications, the present technique shows many advantages; for instance, spectral accuracy, a much higher rate of running, fewer computational hurdles and programming, calculating the tempered-derivative/integral of fractional order, and its spectral accuracy in comparison with other competitive numerical schemes. The study includes stability and convergence analyses and the elapsed times taken to construct the collocation matrices and obtain the numerical solutions, as well as a numerical examination of the produced condition number κ(A) of the resulting linear systems. The accuracy and efficiency of the proposed method are studied from the standpoint of the L2 and L∞-norms error and the fast rate of spectral convergence.

Linear correlated fuzzy solutions of nonlocal problems for multi-order fractional differential systems

This paper introduces the standard Riemann-Liouville-LC fractional derivative D0+βLCRLv(t) for the linear correlated fuzzy-valued function v:J⊂R→RF(E), where β>0, E is a fixed arbitrary fuzzy number. The focus of this study lies in addressing nonlocal problems for multi-order fractional differential systems in the linear correlated fuzzy space RF(E), where the highest order derivative may be greater than one. The solvability of these nonlocal problems is investigated comprehensively, encompassing both cases where E is a symmetric or non-symmetric fuzzy number. The contribution to the calculus literature and the solvability of multi-order fractional differential problems in the fuzzy spaces RF(E) have been presented with some illustrated examples.

New Stability Results for Abstract Fractional Differential Equations with Delay and Non-Instantaneous Impulses

This research delves into the field of fractional differential equations with both non-instantaneous impulses and delay within the framework of Banach spaces. Our objective is to establish adequate conditions that ensure the existence, uniqueness, and Ulam–Hyers–Rassias stability results for our problems. The studied problems encompass abstract impulsive fractional differential problems with finite delay, infinite delay, state-dependent finite delay, and state-dependent infinite delay. To provide clarity and depth, we augment our theoretical results with illustrative examples, illustrating the practical implications of our work.

On Finite-Time Blow-Up Problem for Nonlinear Fractional Reaction Diffusion Equation: Analytical Results and Numerical Simulations

The study of the blow-up phenomenon for fractional reaction–diffusion problems is generally deemed of great importance in dealing with several situations that impact our daily lives, and it is applied in many areas such as finance and economics. In this article, we expand on some previous blow-up results for the explicit values and numerical simulation of finite-time blow-up solutions for a semilinear fractional partial differential problem involving a positive power of the solution. We show the behavior solution of the fractional problem, and the numerical solution of the finite-time blow-up solution is also considered. Finally, some illustrative examples and comparisons with the classical problem with integer order are presented, and the validity of the results is demonstrated.

Numerical method for fractional Advection–Dispersion equation using shifted Vieta–Lucas polynomials

In the pursuit of creating more precise and flexible mathematical models for complex physical phenomena, this study constructs a unique fractional model for the Advection–Dispersion equation. The Advection–Dispersion equation is a fundamental mathematical tool for analyzing fluid dynamics and mass transfer processes, but traditional integer-order models often fail to accurately capture anomalous transport behaviors. To overcome this limitation, we employ a non-singular non-integer operator based on the Heydari–Hosseininia notion. This operator allows us to transform the classical advection–dispersion equation into a more robust and flexible fractional model, which can better represent a variety of transport phenomena across different scales and mediums.The numerical approximation of the proposed fractional Advection–Dispersion equation model is achieved using a set of specific polynomials known as shifted Vieta–Lucas polynomials. Aided by their unique properties, the shifted Vieta–Lucas polynomials enable us to transform the original differential system into a more computationally tractable algebraic system via a non-integer derivative matrix. Furthermore, we undertake a detailed analysis of convergence and truncation error associated with the shifted Vieta–Lucas polynomials, underpinning the validity and stability of our numerical method. To illustrate the efficiency and robustness of the derived method, we provide several example problems which are solved using our method. The corresponding solutions are extensively presented with the aid of figures and tables, demonstrating the method’s remarkable performance in solving the fractional Advection–Dispersion equation model. The proposed method shows promising potential for further application and development in solving other fractional differential equations and related scientific problems.

Nonexistence results for a sequential fractional differential problem

In this paper, we study a class of sequential fractional differential inequalities involving Caputo fractional derivatives with different orders. The nonexistence of nontrivial global solutions is investigated in a suitable space via the test function technique and some properties of fractional integrals. Our results are supported by numerical examples.

Research on Fractional Differential Problem of Two Types of Fractional Analytic Functions

Research on Fractional Differential Problem of Two Types of Fractional Analytic Functions

Fractional Dynamical Systems Solved by a Collocation Method Based on Refinable Spaces

A dynamical system is a particle or set of particles whose state changes over time. The dynamics of the system is described by a set of differential equations. If the derivatives involved are of non-integer order, we obtain a fractional dynamical system. In this paper, we considered a fractional dynamical system with the Caputo fractional derivative. We collocated the fractional differential problem in dyadic nodes and used refinable functions as approximation functions to achieve a good degree of freedom in the choice of the regularity. The collocation method stands out as a particularly useful and attractive tool for solving fractional differential problems of various forms. A numerical result is presented to show that the numerical solution fits the analytical one very well. We collocated the fractional differential problem in dyadic nodes using refinable functions as approximation functions to achieve a good degree of freedom in the choice of regularity.