Fractional Equations Research Articles

An efficient numerical method for two-dimensional fractional integro-differential equations with modified Atangana–Baleanu fractional derivative using operational matrix approach

In the present paper, a numerical method is developed for solving non-homogeneous two-dimensional fractional integro-differential equations with a modified Atangana–Baleanu fractional derivative. The approach is based on the operational matrix method, involving the derivation of operational matrices for the terms of the equation to generate an algebraic system that is solved using mathematical software. First, basic definitions of fractional calculus and block pulse functions are presented. Then, the operational matrices and the method of solution are derived. The existence and uniqueness of the solution of the proposed equation are proved. Examples to test the numerical method are provided, demonstrating the efficiency of the proposed method.

Dynamics of fractional optical solitary waves to the cubic–quintic coupled nonlinear Helmholtz equation

This work investigates the dynamics of optical waves to the generalized coupled nonlinear fractional Helmholtz equation with quintic and cubic nonlinear effects. The evolution of broad multicomponent self-trapping beams in Kerr-type nonlinear media is described by the coupled Helmholtz equation, which also take into account spatial dispersion due to non-paraxial effects. This phenomena is particularly relevant to the progressive miniaturization of optics, where the optical wavelength is similar to the beam width. It is essential to integrate non-Kerr terms, such as the self-steepening and the self frequency shift, into the coupled Helmholtz system in order to investigate the propagation of ultrashort optical pulses in the non-paraxial domain. For optical wave solutions, nonlinear ordinary equation of the governing model is achieved by applying the fractional transformation. The different types of the solutions like bright, dark, singular and mixed type solitons are extracted by applying advanced integration methods, namely modified Sardar subequation method and new Kudryashov method. These solutions provide very useful information on how the system operates. The applied approaches are highly efficient and have significant computational capability to efficiently tackle the nonlinear systems. Additionally, we include a diverse array of graphs to demonstrate the physical interpretation of the obtained solutions in relation to a number of significant parameters, thereby highlighting the impact of fractional derivatives. In the context of the proposed model, these visualizations assist with a comprehensive understanding of the solution’s behavior and characteristics. It is anticipated that these solutions may be significant in the study of wave propagation and related fields.

Fuzzy discrete fractional calculus and fuzzy fractional discrete equations

This paper aims to highlight certain limitations in the study of fuzzy fractional discrete equations (FFDEs) based on the generalized Hukuhara difference (gH-difference) in the previous papers. In general, the equivalence between FFDEs and the associated fuzzy discrete fractional sum equations (FDFSEs) is not achieved, requiring the introduction of an appropriate hypothesis to establish this equivalence. Furthermore, this paper introduces the fundamental theory of fuzzy fractional discrete calculus through granular arithmetic operations between fuzzy intervals to address restrictions in the formerly mentioned approaches involving the generalized Hukuhara difference. These operations are constructed based on the concept of the horizontal membership function (HMF) utilized in multidimensional fuzzy arithmetic (MFA). Additionally, the paper proposes the application of fractional discrete calculus to two types of time-discretization diffusion equations with non-zero right-hand sides. Finally, several numerical examples are provided to validate the main results.

Cancer modeling by fractional derivative equation and chemotherapy stabilizing

This paper discusses the theme of cancer modeling and the problem of control by treatments. We model the cancer cells growth by fractional derivative equation and we stabilize their concentration by chemotherapy. The considered model will be converted into simple partial derivative equation and associated with a viability problem. We use the set-valued analysis to make viable the converted model by regulation law of regulation map. Applied chemotherapy concentration on cancer cells concentration follows the regulation law.

A sparse spectral method for fractional differential equations in one-spatial dimension

We develop a sparse spectral method for a class of fractional differential equations, posed on R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}$$\\end{document}, in one dimension. These equations may include sqrt-Laplacian, Hilbert, derivative, and identity terms. The numerical method utilizes a basis consisting of weighted Chebyshev polynomials of the second kind in conjunction with their Hilbert transforms. The former functions are supported on [-1,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[-1,1]$$\\end{document} whereas the latter have global support. The global approximation space may contain different affine transformations of the basis, mapping [-1,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[-1,1]$$\\end{document} to other intervals. Remarkably, not only are the induced linear systems sparse, but the operator decouples across the different affine transformations. Hence, the solve reduces to solving K independent sparse linear systems of size O(n)×O(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(n)\ imes \\mathcal {O}(n)$$\\end{document}, with O(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(n)$$\\end{document} nonzero entries, where K is the number of different intervals and n is the highest polynomial degree contained in the sum space. This results in an O(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(n)$$\\end{document} complexity solve. Applications to fractional heat and wave equations are considered.

Fractional neutron point kinetics model for reactivity transients of the NuScale and comparison with the classical kinetics approach

In this work, the behavior of the neutron and natural circulation parameters for Reactivity Transients in the PWR reactor called NuScale are analyzed through the novel system equations. It model includes non-integer and ordinary order differentials coupling the neutron point kinetic with the thermos-hydraulic core of the reactor. Transient processes of increase and decrease of the feed flow and the coolant inlet temperature are analyzed to identify the feedback of the thermohydraulic effects with the neutron phenomena in a non-integer order scheme. In the results, it is observed that for short times the power and reactivity have the same behavior with the classical model and the non-integer order model for all values of the anomalous diffusion coefficient simulated. However, for long times of simulation, there is more reactivity and less power, in contrast to the classical model, when reducing the value of the anomalous diffusion coefficient these counter-intuitive effects are increased. Additionally, a delay is observed between the change in reactivity and power. In conclusion, it is observed that sub-diffusive processes are relevant when there are slow changes in reactivity, which can be analyzed through the Fractional Order Neutron Density Equation.

Exact Solutions for the Sharma–Tasso–Olver Equation via the Sardar Subequation Method with a Comparison between Atangana Space–Time Beta-Derivatives and Classical Derivatives

The Sharma–Tasso–Olver (STO) equation is a nonlinear, double-dispersive, partial differential equation that is physically important because it provides insights into the behavior of nonlinear waves and solitons in various physical areas, including fluid dynamics, optical fibers, and plasma physics. In this paper, the STO equation is generalized to a fractional equation by using Atangana (or Atangana–Baleanu) fractional space and time beta-derivatives since they have been found to be useful as a model for a variety of traveling-wave phenomena. Exact solutions are obtained for the integer-order and fractional-order equations by using the Sardar subequation method and an appropriate traveling-wave transformation. The exact solutions are obtained in terms of generalized trigonometric and hyperbolic functions. The exact solutions are derived for the integer-order STO and for a range of values of fractional orders. Numerical solutions are also obtained for a range of parameter values for both the fractional and integer orders to show some of the types of solutions that can occur. As examples, the solutions are obtained showing the physical behavior, such as the solitary wave solutions of the singular kink-type and periodic wave solutions. The results show that the Sardar subequation method provides a straightforward and efficient method for deriving new exact solutions for fractional nonlinear partial differential equations of the STO type.

Numerical solution of nonlinear fractional delay integro-differential equations with convergence analysis

Numerical solution of nonlinear fractional delay integro-differential equations with convergence analysis

Pseudospectral analysis for multidimensional fractional Burgers equation based on Caputo fractional derivative

This study presents the Chebyshev pseudospectral approach in time and space to approximate a solution to the time-fractional multidimensional Burgers equation. The suggested approach utilizes Chebyshev–Gauss–Lobatto (CGL) points in both spatial and temporal directions. To figure out the fractional derivative matrix at CGL points, we use the Caputo fractional derivative formula. Further, the Chebyshev fractional derivative matrix is utilized to reduce the given problem in an algebraic system of equations. The numerical approach known as the Newton–Raphson is implemented to get the desired results for the system. Error analysis for the set of values of ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ u $$\\end{document} is done for various model examples of fractional Burgers equations, where ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document} represents the fractional order. The computed numerical results are in perfect agreement with the exact solutions.

Determining coefficients for a fractional p-Laplace equation from exterior measurements

We consider an inverse problem of determining the coefficients of a fractional p -Laplace equation in the exterior domain. Assuming suitable local regularity of the coefficients in the exterior domain, we offer an explicit reconstruction formula in the region where the exterior measurements are performed. This formula is then used to establish a global uniqueness result for real-analytic coefficients. In addition, we also derive a stability estimate for the unique determination of the coefficients in the exterior measurement set.

Picard and Adomian decomposition methods for a fractional quadratic integral equation via generalized fractional integral

The primary focus of this paper is to thoroughly examine and analyze a class of a fractional quadraticintegral equation via generalized fractional integral. To achieve this, we introduce an operator that possessesfixed points corresponding to the solutions of the fractional quadratic integral equation, effectively transforming thegiven equation into an equivalent fixed-point problem. By applying the Banach fixed-point theorems, we prove theuniqueness of solutions to fractional quadratic integral equation. Additionally, The Adomian decomposition methodis used, to solve the resulting fractional quadratic integral equation. This technique rapidly provides convergentsuccessive approximations of the exact solution to the given fractional quadratic integral equation, therefore, weinvestigate the convergence of approximate solutions, using the Adomian decomposition method. Finally, we providesome examples, to demonstrate our results. Our findings contribute to the current understanding of fractionalquadratic integral equation and their solutions and have the potential to inform future research in this area.

Numerical and Analytical Approaches to Fractional Quantum Mechanics

Our research meticulously navigates the realms of Fractional Quantum Mechanics (FQM), focusing on a critical examination of both numerical and analytical methods that harness the potential of fractional calculus to illuminate the quantum world's complexities. By embarking on this scholarly journey, we aim to decode the intricate dynamics that fractional equations reveal about quantum systems, pushing the boundaries of conventional quantum mechanics. This comparative study meticulously evaluates the efficacy and insights provided by these two distinct approaches, highlighting their contributions to a deeper understanding of quantum phenomena. As we traverse through the layers of quantum dynamics, our work seeks to contribute significantly to the theoretical framework of FQM, offering innovative perspectives and methodologies. The essence of this research lies in its potential to forge new theoretical pathways and inspire further exploration in the quantum domain, leveraging the unique capabilities of fractional calculus. By enriching the scientific community with our findings, we aspire to open new horizons in the study of quantum mechanics, marking a step forward in the ongoing quest to unravel the mysteries of the quantum universe.

An hp-version of the discontinuous Galerkin method for fractional integro-differential equations with weakly singular kernels

An hp-version of the discontinuous Galerkin method for fractional integro-differential equations with weakly singular kernels

Hypercomplex operator calculus for the fractional Helmholtz equation

In this paper, we develop a hypercomplex operator calculus to treat fully analytically boundary value problems for the homogeneous and inhomogeneous fractional Helmholtz equation where fractional derivatives in the sense of Caputo and Riemann–Liouville are applied. Our method extends the recently proposed fractional reduced differential transform method (FRDTM) by using fractional derivatives in all directions. For the special separable case in three dimensions, we obtain completely explicit representations for the fundamental solution. This allows us to interpret and to understand the appearance of spatial steady‐state solutions or spatial blow‐ups of the fractional Helmholtz equation in a better way. More precisely, we were able to present explicit conditions for the parameters in the representation formulas of the fundamental solutions under which we obtain bounded or spatial decreasing steady‐solutions and when spatial blow‐ups occur. We also illustrate this with some representative numerical examples. Furthermore, we show that it is possible to recover the recently studied cases as well as the classical cases as particular limit cases within our more general setting. Using the hypercomplex operator approach also allows us to factorize the fractional Helmholtz operator and obtain some interesting duality relations between left and right derivatives, Caputo and Riemann–Liouville derivatives, and eigensolutions of antipodal eigenvalues in terms of a generalized Borel–Pompeiu formula. This factorization, in turn, allows us to tackle inhomogeneous fractional Helmholtz problems.

On Higher-Order Nonlinear Fractional Elastic Equations with Dependence on Lower Order Derivatives in Nonlinearity

The paper studied high-order nonlinear fractional elastic equations that depend on low-order derivatives in nonlinearity and established the existence and uniqueness results by using the Leray–Schauder alternative theorem and Perov’s fixed point theorem on an appropriate space under mild assumptions. Examples are given to illustrate the key results.

Galerkin spectral and finite difference methods for the solution of fourth-order time fractional partial integro-differential equation with a weakly singular kernel

Galerkin spectral and finite difference methods for the solution of fourth-order time fractional partial integro-differential equation with a weakly singular kernel

Bifurcation analysis and soliton structures of the truncated [formula omitted]-fractional Kuralay-II equation with two analytical techniques

The truncated M-fractional Kuralay (TMFK)-II equation is prevalent in the exploration of specific complex nonlinear wave phenomena. Such types of wave phenomena are more applicable in science and engineering. These equations could potentially provide insights into understanding the intricate dynamics of optical phenomena, encompassing solitons, nonlinear effects, and wave interactions. This study aims to uncover a diverse range of soliton solutions to the model, spanning trigonometric, hyperbolic, exponential, and rational expressions. These solutions are unveiled through the application of extended hyperbolic functions and improved F-expansion techniques, representing the primary objective of this research. The three-dimensional (3D) and two-dimensional (2D) combined charts are plotted for some of these solutions. The impact of the fractional parameters and time variations is also illustrated. Moreover, the models are converted into a planar dynamical system using a Galilean transformation, and the analysis of bifurcation is examined. This research underscores the versatility of the aforementioned techniques for exploring complex nonlinear phenomena across various engineering and scientific disciplines. Finally, the findings of this study hold significant implications for advancing our understanding and analysis of nonlinear wave dynamics in diverse physical systems.

Atomic Solution for Fractional Non-linear Burgers Equation of Conformable Type

Atomic Solution for Fractional Non-linear Burgers Equation of Conformable Type

Numerical approximation of a generalized time fractional partial integro-differential equation of Volterra type based on a meshless method

The moving least squares (MLS) approximation is used in this study to solve time-fractional partial integro-differential equations (TFPIDE). In our approach, we approximate the time Caputo fractional derivative term and the integral term by using a finite difference scheme and trapezoidal method, respectively, which yields an order of accuracy of O(τ+τ2−α), where 0<α<1. We thoroughly investigate the applicability and validity of the proposed method and provide an error estimate for its performance. Additionally, we solve various numerical problems to demonstrate the accuracy and computational efficiency of our approach, confirming the theoretical findings.

A necessary and sufficient conditions for the global existence of solutions to fractional reaction-diffusion equations on $$\mathbb {R}^{N}$$

A necessary and sufficient conditions for the global existence of solutions to fractional reaction-diffusion equations on $$\mathbb {R}^{N}$$