Fractional Differential Equations Research Articles

Fractional calculus for distributions

AbstractFractional derivatives and integrals for measures and distributions are reviewed. The focus is on domains and co-domains for translation invariant fractional operators. Fractional derivatives and integrals interpreted as "Equation missing"-convolution operators with power law kernels are found to have the largest domains of definition. As a result, extending domains from functions to distributions via convolution operators contributes to far reaching unifications of many previously existing definitions of fractional integrals and derivatives. Weyl fractional operators are thereby extended to distributions using the method of adjoints. In addition, discretized fractional calculus and fractional calculus of periodic distributions can both be formulated and understood in terms of "Equation missing"-convolution.

Study of a Coupled Ψ–Liouville–Riemann Fractional Differential System Characterized by Mixed Boundary Conditions

Study of a Coupled Ψ–Liouville–Riemann Fractional Differential System Characterized by Mixed Boundary Conditions

Extensions of Bicomplex Hypergeometric Functions and Riemann–Liouville Fractional Calculus in Bicomplex Numbers

In this paper, we present novel advancements in the theory of bicomplex hypergeometric functions and their applications. We extend the hypergeometric function to bicomplex parameters, analyse its convergence region, and define its integral and derivative representations. Furthermore, we delve into the k-Riemann–Liouville fractional integral and derivative within a bicomplex operator, proving several significant theorems. The developed bicomplex hypergeometric functions and bicomplex fractional operators are demonstrated to have practical applications in various fields. This paper also highlights the essential concepts and properties of bicomplex numbers, special functions, and fractional calculus. Our results enhance the overall comprehension and possible applications of bicomplex numbers in mathematical analysis and applied sciences, providing a solid foundation for future research in this field.

Ulam–Hyers–Rassias Mittag-Leffler stability of ϖ–fractional partial differential equations

This paper offers a comprehensive analysis of solution representations for ϖ-fractional partial differential equations, specifically focusing on the linear case of the Darboux problem. We exhibit a representation of the solutions for the Darboux problem of ϖ-fractional partial differential equations in the linear case in the space of continuous functions. Through the application of the generalized Gronwall inequality, we establish the Ulam–Hyers–Rassias Mittag–Leffler stability in the space of continuous functions. Three numerical examples are presented to show the effectiveness and the applicability of our results.

A polynomial collocation method for a class of singular fractional differential equations

In this work we consider a class of singular fractional differential equations (SFDEs). Using a suitable variable transformation we rewrite the SFDE as a cordial Volterra integral equation and propose a polynomial collocation method to find an approximate solution of the original problem. We provide the error analysis of the numerical method and we illustrate its feasibility and accuracy through some numerical examples.

Magnetized electroosmotic drug delivery of Au-SiO2 nanocarriers through blood-based jeffrey hybrid nanofluid in a microchannel: Caputo Fabrizio derivative modeling

Fractional calculus is emerging as a promising field to overcome the intricacies inherent in biological systems that prevent conventional techniques from producing optimal results. The present research emphasizes the impact of thermal radiation, chemical reactions, and radiation absorption on an electroosmotic magnetohydrodynamic (MHD) blood-based Jeffrey hybrid nanofluid flow in a microchannel, employing the novel Caputo-Fabrizio fractional calculus approach. This study is carried out on two models: ramped and constant boundary conditions with distinct zeta potentials. The graphs are drawn with the help of MATLAB software. Our results demonstrate that the fractional order significantly influences the drug dispersion, and for α=0.9 (fractional parameter), the blood flow becomes wavy. The presence of nanoparticles improves drug transport, hence enhancing the drug concentration in proximity to the target site. For α=0.1, the Jeffrey nanofluid with ramped conditions shows the highest velocity enhancement in the case of pressure-driven, natural convective flow. Electroosmotic force facilitates fluid flow and enhances drug transport efficiency. For Ha < 4, the blood velocity decreases in the vicinity of the plate y = 0, and reverse behavior is observed as it passes y = 0.5, which can aid in effective drug delivery. At y = 0, the heat transfer rate increases by a maximum of 199.18 % while skin friction decreases to 3.07 %, aiding in maintaining medications at the desired temperature and improving drug delivery efficiency. The temperature and velocity of the blood hybrid nanofluid are maximized under ramping wall settings compared to constant wall conditions.

Exact wave solutions of truncated M-fractional Boussinesq-Burgers system via an effective method

Abstract In this paper, we present distinct types of exact wave soliton solutions of an important fluid flow dynamic system called the truncated M-fractional (1+1)-dimensional nonlinear Boussinesq-Burgers system (BBS). This model is used to explain ocean waves, matter-wave pulses, waves in ferromagnetic media, the proliferation of waves in shallow water, etc. We transform the nonlinear fractional system into a nonlinear ordinary differential equation by using a fractional transformation to obtain dark, bright, singular, dark-bright, dark-singular, bright-singular and periodic type solitons solutions by employing the modified extended tanh function method (METhFM). The use of fractional derivatives makes the solutions different from the existing solutions. The obtained results are useful in the optical fibers, fluid dynamics, ocean engineering and other related fields. To visualize the system’s behavior, some of the solutions are represented by two- and three-dimensional graphs which are obtained and verified with the help of Mathematica. The achieved results provide a better understanding of the behavior of the nonlinear fractional partial differential equations and the dynamics of BBS, which are not present in the literature and are helpful in future studies of the concerned system.

Fractional Differential Equations with Impulsive Effects

This paper discusses impulsive effects on fractional differential equations. Two approaches are taken to obtain our results: either with fixed or changing lower limits in Caputo fractional derivatives. First, we derive an existence result for periodic solutions of fractional differential equations with periodically changing lower limits. Then, the impulsive effects are modeled for fractional differential equations regarding the nonlinearities rather than the initial value conditions. The proposed impulsive model differs from common discontinuous and nonsmooth dynamical systems.

A New Perspective for Scientific Modelling: Sparse Reconstruction Based Approach for Learning Time-Space Fractional Differential Equations

Abstract This paper studies a sparse reconstruction based approach to learn time-space fractional differential equations, i.e. to identify parameter values and particularly the order of the fractional derivatives. The approach uses a generalized Taylor series expansion to generate, in every iteration, a feature matrix which is used to learn the fractional orders of both, temporal and spatial derivatives by minimizing the LASSO operator using Differential Evolution algorithm. To verify the robustness of the method, numerical results for time-space fractional diffusion equation, wave equation, and Burgers? equation at different noise levels in the data are presented. Finally, the methodology is applied to a realistic example where underlying fractional differential equation associated to published experimental data obtained from an in-vitro cell culture assay is learned.

An extension of Schweitzer's inequality to Riemann-Liouville fractional integral

Abstract This note focuses on establishing a fractional version akin to the Schweitzer inequality, specifically tailored to accommodate the left-sided Riemann-Liouville fractional integral operator. The Schweitzer inequality is a fundamental mathematical expression, and extending it to the fractional realm holds significance in advancing our understanding and applications of fractional calculus.

Optimizing Fractional Differential Equation Solutions with Novel Müntz Space Basis Functions

This paper introduces innovative basis functions derived from Müntz spaces, aimed at addressing the computational challenges of Fractional Differential Equations (FDEs). Our primary focus is the creation of these functions using singular indices linked to the solutions of FDEs. We thoroughly investigate the properties of these fundamental functions to understand their operational potential. These functions are particularly adept at capturing initial singular indices, making them highly suitable for solving FDEs. The proposed numerical method is distinguished by its rapid convergence rates, showcasing its efficiency in computational evaluations. We validate our approach by presenting numerical examples that highlight its accuracy and reliability. These examples confirm the effectiveness and efficiency of the new basis functions from Müntz spaces in accurately solving FDEs. This research advances numerical methods for FDEs and serves as a valuable resource for researchers seeking robust and reliable techniques

Soham Transform in Fractional Differential Equations

Soham Transform in Fractional Differential Equations

Approximate Controllability of Hilfer Fractional Stochastic Evolution Inclusions of Order 1 &lt; q &lt; 2

This paper addresses the approximate controllability results for Hilfer fractional stochastic differential inclusions of order 1&lt;q&lt;2. Stochastic analysis, cosine families, fixed point theory, and fractional calculus provide the foundation of the main results. First, we explored the prospects of finding mild solutions for the Hilfer fractional stochastic differential equation. Subsequently, we determined that the specified system is approximately controllable. Finally, an example displays the theoretical application of the results.

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.

Operational Calculus for the 1st-Level General Fractional Derivatives and Its Applications

The 1st-level General Fractional Derivatives (GFDs) combine in one definition the GFDs of the Riemann–Liouville type and the regularized GFDs (or the GFDs of the Caputo type) that have been recently introduced and actively studied in the fractional calculus literature. In this paper, we first construct an operational calculus of the Mikusiński type for the 1st-level GFDs. In particular, it includes the operational calculi for the GFDs of the Riemann–Liouville type and for the regularized GFDs as its particular cases. In the second part of the paper, this calculus is applied for the derivation of the closed-form solution formulas to the initial-value problems for the linear fractional differential equations with the 1st-level GFDs.

System of Nonlinear (&lt;i&gt;k&lt;/i&gt;, ψ)-Hilfer Fractional Order Hybrid Boundary Value Problems

This paper focuses on the existence and uniqueness of solutions for a new class of coupled fractional differential equations subject to specific boundary conditions. We introduce a novel approach that incorporates the (k, ψ)-Hilfer fractional derivative, a recent advancement in fractional calculus. By leveraging the properties of this operator, we develop a comprehensive framework to analyze the proposed system. Our findings, established through the application of Schauder's fixed point theorem and the Banach contraction principle, contribute to the understanding of complex phenomena modeled by fractional differential equations. This research expands the boundaries of fractional calculus and offers potential applications in various scientific and engineering domains.

Shifted Legendre Gauss‐Lobatto collocation scheme for solving nonlinear coupled space‐time fractional reaction‐advection diffusion equations

AbstractThis work aims to develop and apply an algorithm for solving space‐time fractional reaction‐advection diffusion (STFRAD) nonlinear coupled equations with specified initial and boundary conditions by employing a Shifted Legendre Gauss‐Lobatto collocation (SLGLC) scheme. The fractional derivatives of time and space are defined in Caputo sense. The approximate solutions have been constructed with the help of shifted Legendre polynomials. The proposed numerical scheme reduces the coupled fractional order differential equations into a system of algebraic equations which can then be easily solved. The error analysis through the two test problems is carried out to validate the efficiency and efficacy of the method. The effect of advection and reaction terms, and various space‐time fractional order derivatives on the solution profiles have been studied and are shown graphically for different particular cases. It has been observed that the species whose transport is simulated using STFRADE shows higher concentration at a specific distance in porous media in case of integer order differential equations as compared to fractional order equations demonstrating overestimation of its impact in case of integer order differential equation. This is extremely important observation with respect to applications in the field of groundwater pollution management. The salient feature of the article is the stability and convergence analyses of the proposed scheme on the concerned model, which demonstrates the effectiveness and capabilities of the developed method.

Application of the Sumudu Variational Iteration Method with Atangana-Baleanu-Caputo Operator for Solving Fractional-Order Heat-Like Equations with Initial Conditions

Fractional calculus techniques are widely utilized across various engineering disciplines and applied sciences. Among these techniques is the Sumudu Variational Iteration Method (SVIM), which has not yet been tested with the Atangana-Baleanu-Caputo fractional derivative in academic literature. This work aims to explore the application of SVIM for solving fractional-order partial differential equations using the Atangana-Baleanu-Caputo derivative. The method integrates the Sumudu transform with the variational iteration method. To demonstrate the effectiveness and validity of SVIM, we apply it to solve one-dimensional (1-D), two-dimensional (2-D), and three-dimensional (3-D) fractional-order heat-like partial differential equations. The results indicate that SVIM is both convergent and efficient for solving these types of fractional partial differential equations.

Existence of solution for some nonlinear g-Caputo fractional-order differential equations based on Wardowski–Mizoguchi–Takahashi contractions

In this study, we prove the existence and uniqueness of a solution to a g-Caputo fractional differential equation with new boundary value conditions utilizing the combined Wardowski–Mizoguchi–Takahashi contractions via reduction of this equation to a fractional integral equation. We provide an example to demonstrate our findings.

A fractional calculus approach to smoking dynamics with bifurcation analysis

A fractional calculus approach to smoking dynamics with bifurcation analysis