Fractional Operator Research Articles

A convergence result for the derivation of front propagation in nonlocal phase field models

We prove that the mean curvature of a smooth surface in \R^{n} , n\geq 2 , arises as the limit of a sequence of functions that are intrinsically related to the difference between an n - and 1 -dimensional fractional Laplacian of a phase transition. Depending on the order of the fractional Laplace operator, we recover the fractional mean curvature or the classical mean curvature of the surface. Moreover, we show that this is an essential ingredient for deriving the evolution of fronts in fractional reaction-diffusion equations such as those for atomic dislocations in crystals.

Discretization of Fractional Fully Nonlinear Equations by Powers of Discrete Laplacians

AbstractWe study discretizations of fractional fully nonlinear equations by powers of discrete Laplacians. Our problems are parabolic and of order $$\sigma \in (0,2)$$ σ ∈ ( 0 , 2 ) since they involve fractional Laplace operators $$(-\Delta )^{\sigma /2}$$ ( - Δ ) σ / 2 . They arise e.g. in control and game theory as dynamic programming equations – HJB and Isaacs equation – and solutions are non-smooth in general and should be interpreted as viscosity solutions. Our approximations are realized as finite-difference quadrature approximations and are 2nd order accurate for all values of $$\sigma $$ σ . The accuracy of previous approximations of fractional fully nonlinear equations depend on $$\sigma $$ σ and are worse when $$\sigma $$ σ is close to 2. We show that the schemes are monotone, consistent, $$L^\infty $$ L ∞ -stable, and convergent using a priori estimates, viscosity solutions theory, and the method of half-relaxed limits. We also prove a second order error bound for smooth solutions and present many numerical examples.

Study of fractional order rabies transmission model via Atangana–Baleanu derivative

In this work, we aim at disentangling the theoretical contribution through mathematical modeling approach to advance the understanding of rabies dynamics and control in livestock population. A fractional order model of rabies, using Atangana–Baleanu fractional operator is created. The analysis of suggested system and its application are both conducted. The capacity of proposed model to forecast the disease can help researchers and livestock health care agencies to take preventive actions.

ANALYSIS OF STUDENTS' UNDERSTANDING OF MATHEMATICAL CONCEPTS ON FRACTIONS IN GRADE VIII AT SMP NEGERI 2 MAZINO

This study aims to describe and identify the factors contributing to the low understanding of mathematical concepts related to fractions among students. This research is qualitative in nature with a descriptive approach. The informants in this study are 18 eighth-grade students from SMP Negeri 2 Mazino. Data were analyzed through the steps of data reduction, data presentation, and conclusion drawing. The data collection techniques used are tests and interviews. The results show that, overall, students exhibit low understanding of concepts when distinguishing between two fractions, with 18% categorized as very poor. In classifying fractions by type, the percentage is 19% in the very poor category. For ordering fractions from smallest to largest, the percentage is 37%, categorized as poor. In simplifying addition and subtraction operations involving fractions, 22% fall into the poor category. Additionally, in relating everyday life to fraction concepts, the percentage is 60%, categorized as sufficient, while 20% fall into the very poor category. Factors influencing this include a lack of understanding of the statements in the tests, not comprehending the questions posed, being less meticulous in understanding simple fractions, and difficulty in interpreting word problems related to fraction operations. In conclusion, eighth-grade students at SMP Negeri 2 Mazino demonstrate low understanding in completing tests on fractions. It is recommended that teachers create a question-and-answer space in their lesson plans to enhance students' conceptual understanding when working with fractions.

Does a Fractional-Order Recurrent Neural Network Improve the Identification of Chaotic Dynamics?

This paper presents a quantitative study of the effects of using arbitrary-order operators in Neural Networks. It is based on a Recurrent Wavelet First-Order Neural Network (RWFONN), which can accurately identify several chaotic systems (measured by the mean square error and the coefficient of determination, also known as R-Squared, r2) under a fixed parameter scheme in the neural algorithm. Using fractional operators, we analyze whether the identification capabilities of the RWFONN are improved, and whether it can identify signals from fractional-order chaotic systems. The results presented in this paper show that using a fractional-order Neural Network does not bring significant advantages in the identification process, compared to an integer-order RWFONN. Nevertheless, the neural algorithm (modeled with an integer-order derivative) proved capable of identifying fractional-order dynamical systems, whose behavior ranges from periodic and multi-stable to chaotic oscillations. That is, the performances of the Neural Network model with an integer-order derivative and the fractional-order network are practically identical, making the use of fractional-order RWFONN-type networks meaningless. The results deepen the work previously published by the authors, and contribute to developing structures based on robust and generic neural algorithms to identify more than one chaotic oscillator without retraining the Neural Network.

An Error Analysis Study for the Distributed-Order Time-Fraction Model Using an Efficient Numerical Method

In this paper, we propose a numerical scheme based on the shifted Legendre polynomials for solving the forced Korteweg–de Vries (fKdV) equation including a Caputo fractional operator of a distributed order. To obtain numerical solutions of these types of equations, we derive an operational matrix based on the shifted Legendre polynomials, and using this operational matrix, their equations change to a set of nonlinear algebraic systems. Then, by calculating these systems in the collocation points, we solve systems. Also, convergence and error are investigated in this paper. Finally, several numerical examples to show the applicability of our scheme are displayed.

Chaos and feasibility analysis of a healthy lifestyle’s sustainability using fractional-order polycystic ovarian syndrome’s effect on infertility

In this study, we developed a fractional-order model for polycystic ovarian syndrome issue transmission in women through four compartments with the Mittag-Leffler kernel. For biological feasibility, we discussed the positivity and boundedness of solutions with fixed point mappings in Banach spaces. Sensitivity analysis is performed using different variables as response functions each time, employing Latin Hyper-cube Sampling (LHS) and Partial Rank Correlation Coefficient (PRCC) with 200 iterations. Further, the reproductive number is determined for the model’s biological feasibility with an analysis of parameter impact through Matlab software. The solutions are examined for global stability using the Lyapunov function with a first derivative. Chaos Control will use the regulate for linear responses approach to bring the system to stabilize according to its points of equilibrium. Numerical solutions are derived by using a Newton polynomial in the generalized form of the Mittag-Leffler kernel, which illustrates the effect of the fractional operator and parameters involved in the system. Results are also analyzed with different assumptions of parameter values and initial conditions to observe the different impacts of treatment to overcome the risk of infertility in women. Such type of investigation will be useful to investigate the spread of disease as well as helpful in developing control strategies to overcome the risk of problems and best management care pre-detection of the issue and for treatment.

Existence and uniqueness of the solution to initial and inverse problems for integro-differential heat equations with fractional load

This work is devoted to the unique solvability of the direct and inverse problems for a multidimensional heat equation with a fractional load in Holder spaces. In the problem under consideration, the loaded term is in the form of a fractional integral operator for the time variable. We prove the existence and uniqueness of the solution to these problems by the contraction mapping theorem and the theory of integral equations. For more information see https://ejde.math.txstate.edu/Volumes/2024/64/abstr.html

Modeling and analysis using piecewise hybrid fractional operator in time scale measure for ebola virus epidemics under Mittag–Leffler kernel

This emerging infectious disease poses one of the most severe threats to public health in these locations, but there are not many reliable therapies yet. In this work, we developed the Ebola virus dynamics and control factors epidemic model with a piecewise hybrid fractional Operator in time scale measure insight of Mittag–Leffler kernel. Patterns and structures that repeat at various scales are the focus of fractal analysis, which has applications in complex systems such as biological ones. Both qualitatively and statistically, a proposed model with the Lipschitz criteria and linear growth is examined, considering positive solutions, boundedness, and uniqueness at equilibrium points with Leray-Schauder results under time scale ideas. The regulation for linear responses approach will be used by Chaos Control to stabilize the system after its equilibrium points. A fractional-order framework with a controlled design will be considered, where solutions are bounded in the feasible domain of relations of different compartments. Ulam–Hyers stability results in the solution are treated when function (constant or rising) for the component of qualitative inquiry in generalized form. The dynamical behaviors of the suggested model are discussed with the Newton polynomial approach used to implement on model in the sense of classical piecewise and Mittag–Leffler kernel at different fractional order values. The model shows that solutions are stable and confined within a feasible range, ensuring reliability. Through detailed simulations, it effectively captures how different interventions and infection rates influence Ebola spread. This fractional-order model enhances understanding of Ebola transmission, providing a strong basis for predicting outbreaks and planning effective control measures, with practical applications for analyzing real-world data.

Dynamics behaviours of N-kink solitons in conformable Fisher–Kolmogorov–Petrovskii–Piskunov equation

PurposeThis manuscript is related to compute $N$-kink soliton solutions for conformable Fisher–Kolmogorov equation (CFKE) by using the generalized extended direct algebraic method (EDAM). The considered problem has important applications in mathematical biology and reaction diffusion processes. Also, the mentioned problem has significant applications in population dynamics. The fractional order conformable derivative has many features as compared to the other fractional order differential operators. For instance, the chain, product and quotient procedures do not satisfy by other fractional differential operators, but conformable operators obey the mentioned rules. Hence, we compute the soliton solutions for the mentioned problem and present its various dynamical behaviours graphically.Design/methodology/approachThe generalized EDAM is used in this article to examine the calculation of N-kink soliton solutions for the CFKE. In mathematical biology and reaction-diffusion processes, the topic under consideration holds great significance, especially when considering population dynamics.FindingsThe results highlight the benefits of utilising conformable derivatives in mathematical modelling and further our understanding of fractional differential equations and their applications.Research limitations/implicationsThe work focuses primarily on N-kink soliton solutions, which may limit the examination of alternative types of solutions (e.g., multi-soliton or periodic solutions) that might give new insights into the dynamics of the CFKE.Practical implicationsThe generated N N-kink soliton solutions can enhance mathematical models in biological contexts, notably in modelling population dynamics, disease propagation and ecological interactions, leading to better forecasts and interventions.Social implicationsPublic health initiatives can benefit from the understanding of disease transmission and intervention efficacy that comes from modelling population dynamics and reaction-diffusion processes.Originality/valueThe use of the generalized EDAM to obtain solutions for N-kink soliton problems is an innovative method for solving the conformable Fisher–Kolmogorov equation, demonstrating the power of this mathematical tool.

A Novel and Accurate Algorithm for Solving Fractional Diffusion-Wave Equations

The main objective of this work is to apply a novel and accurate algorithm for solving the second-order and fourth-order fractional diffusion-wave equations (FDWEs). First, the desired equation is reduced to the corresponding Volterra integral equation (VIE). Then, the collocation method is applied, for which the Chebyshev cardinal functions (CCFs) have been considered as the bases. In this paper, the CCFs based on a Lobatto grid are introduced and used for the first time to solve these kinds of equations. To this end, the derivative and fractional integral operators are represented in CCFs. The main features of the method are simplicity, compliance with boundary conditions, and good accuracy. An exact analysis to show the convergence of the scheme is presented, and illustrative examples confirm our investigation.

Fractional integral related to Schrödinger operator on vanishing generalized mixed Morrey spaces

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The fractional integral operator associated with L is denoted by IβL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\mathcal{I}}_{\\beta}^{L}$\\end{document}. We investigate the boundedness of IβL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\mathcal{I}}_{\\beta}^{L}$\\end{document} and [b,IβL]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$[b,{\\mathcal{I}}_{\\beta}^{L}]$\\end{document}, which are its commutators with bθ(ρ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$b_{\ heta}(\\rho )$\\end{document} on vanishing generalized mixed Morrey spaces VMp→,φα,V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$VM_{\\vec{p},\\varphi}^{\\alpha ,V}$\\end{document} related to Schrödinger operation and generalized mixed Morrey spaces Mp→,φα,V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$M_{\\vec{p},\\varphi}^{\\alpha ,V}$\\end{document}. The boundedness of the operator IβL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\mathcal{I}}_{\\beta}^{L}$\\end{document} is ensured by finding sufficient conditions on the pair (φ1,φ2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\varphi _{1},\\varphi _{2})$\\end{document}, which goes from Mp→,φ1α,V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$M_{\\vec{p},\\varphi _{1}}^{\\alpha ,V}$\\end{document} to Mq→,φ2α,V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$M_{\\vec{q},\\varphi _{2}}^{\\alpha ,V}$\\end{document}, and from VMp→,φ1α,V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$VM_{\\vec{p},\\varphi _{1}}^{\\alpha ,V}$\\end{document} to VMq→,φ2α,V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$VM_{\\vec{q},\\varphi _{2}}^{\\alpha ,V}$\\end{document}, ∑i=1n1pi−∑i=1n1qi=β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\sum \\limits _{i=1}^{n}\\frac{1}{p_{i}}-\\sum \\limits _{i=1}^{n}\\frac{1}{q_{i}}=\\beta $\\end{document}. When b belongs to BMOθ(ρ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$BMO_{\ heta}(\\rho )$\\end{document} and (φ1,φ2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\varphi _{1},\\varphi _{2})$\\end{document} satisfies some conditions, we also show that the commutator operator [b,IβL]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$[b,{\\mathcal{I}}_{\\beta}^{L}]$\\end{document} is bounded from Mp→,φ1α,V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$M_{\\vec{p},\\varphi _{1}}^{\\alpha ,V}$\\end{document} to Mq→,φ2α,V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$M_{\\vec{q},\\varphi _{2}}^{\\alpha ,V}$\\end{document} and from VMp→,φ1α,V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$VM_{\\vec{p},\\varphi _{1}}^{\\alpha ,V}$\\end{document} to VMq→,φ2α,V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$VM_{\\vec{q},\\varphi _{2}}^{\\alpha ,V}$\\end{document}.

On bivariate fractal interpolation for countable data and associated nonlinear fractal operator

Abstract Fractal interpolation has been conventionally treated as a method to construct a univariate continuous function interpolating a given finite data set with the distinguishing property that the graph of the interpolating function is the attractor of a suitable iterated function system. On the one hand, attempts have been made to extend the univariate fractal interpolation from a finite data set to a countably infinite set. On the other hand, fractal interpolation in higher dimensions, particularly the theory of fractal interpolation surfaces (FISs), has received increasing attention for more than a quarter century. This article targets a two-fold extension of the notion of fractal interpolation by providing a general framework to construct FISs for a prescribed set consisting of countably infinite data on a rectangular grid. By using this as a crucial tool, we obtain a parameterized family of bivariate fractal functions simultaneously interpolating and approximating a prescribed bivariate continuous function. Some elementary properties of the associated nonlinear (not necessarily linear) fractal operators are established, thereby allowing the interaction of the notion of fractal interpolation with the theory of nonlinear operators.

Endpoint estimates for commutators with respect to the fractional integral operators on Orlicz–Morrey spaces

Endpoint estimates for commutators with respect to the fractional integral operators on Orlicz–Morrey spaces

A NEW FORMULATION AND ANALYTICAL APPLICATIONS OF FRACTIONAL OPERATORS

This research paper introduces a novel formulation of the modified Atangana–Baleanu (AB) Fractional Operators (FrOs). The paper begins by discussing the boundedness of the novel fractional derivative operator. Some fractional differential equations corresponding to different choices of functions as well as comparative graphical representations of a function and its derivative are provided. Furthermore, the paper investigates the generalized Laplace transform for this newly introduced operator. By employing the generalized Laplace transform, a wide range of fractional differential equations can be effectively solved. Additionally, the paper establishes the corresponding form of the AB Caputo fractional integral operator, examines its boundedness and obtains its Laplace transform. It is worth noting that the FrOs previously documented in the existing literature can be derived as special cases of these recently explored FrOs.

On proportional hybrid operators in the discrete setting

In this article, we introduce a new nonlocal operator defined as a linear combination of the discrete fractional Caputo operator and the fractional sum operator. A new dual operator is also introduced by replacing the discrete fractional Caputo operator with the discrete fractional Riemann–Liouville operator. It is shown that it corresponds to a natural discretization of a proportional hybrid operator defined by the Riemann–Liouville operator instead of Caputo hybrid operator. We then analyze the most important properties of these operators, such as their inverse operator and the ‐transform, among others. As an application, we solve difference equations equipped with these operators and obtain explicit solutions for them in terms of trivariate Mittag‐Leffler sequences.

The impact of the Caputo fractional difference operator on the dynamical behavior of a discrete-time SIR model for influenza A virus

Abstract Since there are few studies that deal with the fractional-order discrete-time epidemic models, this paper presents a new fractional-order discrete-time SIR epidemic model that is constructed based on the Caputo fractional difference operator. The effect of the fractional orders on the global dynamics of the SIR model is analyzed. In particular, the existence and stability of equilibrium points of the model are presented. Furthermore, we investigate the qualitative dynamical properties of the SIR model for both commensurate and incommensurate fractional orders using powerful nonlinear tools such as phase attractors, bifurcation diagrams, maximum Lyapunov exponent, chaos diagrams, and 0-1 test. In addition, the complexity of the discrete model is measured via the spectral entropy complexity algorithm. Further, an active controller is designed to stabilize the chaotic dynamics of the fractional-order SIR model. Finally, the suggested model is fitted with real data to show the accuracy of the current stability study. Our goal was achieved by confirming that the proposed SIR model can display a variety of epidiomologically observed states, including stable, periodic, and chaotic behaviors. The findings suggest that any change in parameter values or fractional orders could lead to unpredictable behavior. As a result, there is a need for additional research on this topic.

Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces

Abstract In this work, we present necessary and sufficient conditions for the boundedness of the commutators generated by multilinear fractional maximal operators on the products of Morrey spaces when the symbol belongs to Lipschitz spaces.

Enhancing Learning to Solve Multicomponent Fractional Viscoelastic Equations with U-net Fourier Neural Operators

Abstract The research of viscoelastic media is currently a hot topic in the interpretation and processing of seismic data. To accurately simulate the propagation of seismic waves in viscoelastic media, the fractional viscoelastic equation has emerged as an indispensable method. However, solving this equation numerically has proven to be challenging due to the complexity introduced by its fractional Laplacian operators. Recently, deep learning, especially Fourier neural operators (FNO), has shown excellent performance in learning to fast solve partial differential equations. Traditional FNO methods may face crosstalk problem and make it difficult to achieve satisfactory accuracy, when solving the multi-component fractional order viscoelastic equation. To solve this problem, we introduce a novel approach based on U-net Fourier neural operator (U-FNO). As an enhanced learning method to the traditional FNO-based method, the U-FNO-based method integrates a U-Fourier layer following the standard Fourier layer as a form of regularization, thereby achieving superior prediction accuracy for multi-component equations. Specifically, both the Fourier layers and U-Fourier layers in U-FNO are trained with the solutions of the equation from previous time steps as inputs. This training process enables the U-FNO to efficiently produce more accurate solutions for subsequent wavefield. Numerical simulations reveal that the U-FNO-based method efficiently learns to solve the fractional viscoelastic wave equation independent of fractional Laplacian operators. Additionally, U-FNO-based method offers superior prediction accuracy in comparison with the traditional FNO-based method.

Mathematical Evaluation and Dynamic Transmissions of a Tumor Growth Model Using a Generalized Singular and Non-Local Kernel

Currently, fractional calculus plays a critical role in improving control techniques, analyzing disease transmission dynamics, and solving several other real-world problems. This research investigates the time-fractional tumor growth model using an innovative approach. The new modified fractional derivative operator employs a singular and non-local kernel, based on Atangana and Baleanu's concepts with the Caputo derivative. The tumor growth model used the newly modified fractional operator, which provided numerical simulation. With the introduction of this new operator, we provide significant analysis for the tumor growth epidemic model. We have proven the uniqueness and stability conditions of the model by utilizing Banach's fixed point theory and the Picard successive approximation method. Using the Laplace-Adomian decomposition method (LADM), we found the numerical solution to the Modified Atangana-Baleanu-Caputo derivative model. We have verified the convergence analysis of the suggested scheme. We ultimately utilize the suggested method to obtain numeric outcomes and simulations for the tumor growth model. The study investigates the effect of multiple biological variables on the transmission of tumor growth dynamics.