Fractional Differential Equations Research Articles

Mathematical Exploration of Malaria Transmission Dynamics: Insights from Fractional Models and Numerical Simulation

AbstractThis study presents an innovative mathematical model denoted as the fractional SIP(H)–SI(M) model, which aims to analyze and understand the dynamics of malaria transmission and spread. This model is distinguished by incorporating memory effects through fractional differential equations, allowing for a more accurate and realistic analysis of disease spread compared to traditional models. The proposed model is applied to Algeria by estimating its parameters using recent health data (from 2000). The results revealed that the disease‐free equilibrium is stable only when the basic reproduction number is less than one, indicating that controlling the spread of malaria and possibly eradicating it can be achieved by implementing appropriate preventive measures. Simulations also demonstrated a direct correlation between the rate of infection transmission and an increase in the number of infected individuals, highlighting the need for swift action when signs of an outbreak emerge. Based on these findings, a set of preventive measures is recommended, including insecticide spraying programs, widespread distribution of insecticide‐treated bed nets, and implementation of effective treatment protocols for infected individuals. This study also emphasizes the importance of continuous monitoring of health data and updating model parameters to ensure the effectiveness and sustainability of preventive measures.

Operational matrix method approach for fractional partial differential-equations

Abstract Fractional partial differential equations (FPDEs) have become very popular to model and analyze numerous different physical phenomena in recent years. However, it is generally complicated to find the exact solutions of those FPDEs. The objective of this study is to find the approximate numerical solution of FPDEs by introducing a wavelet-based operational matrix technique. In this study we employ Hermite wavelets (HWs) and the operational matrices of the fractional integration for Hermite wavelets. The sparsity of the obtained operational matrices provides fast and efficient computation of the proposed method. The original FPDE equations are converted to Sylvester equations, which then are solved to obtain the final solution. We provide a few numerical examples to demonstrate the versatility and efficiency of the proposed method.

A Special Note According to Possible Applications of Fractional-Order Calculus for Various Special Functions

The main aim of this special study is to recall certain information about fractional (arbitrary) order calculus, which has wide and fruitful applications in science and engineering. Then, it aims to consider various essential definitions related to fractional order integrals and derivatives for stating and proving some results, as well as to present some of their possible applications to the attention of related researchers.

On the Global Existence of Solution and Lyapunov Asymptotic Practical Stability for Nonlinear Impulsive Caputo Fractional Derivative via Comparison Principle

This paper examines the existence of maximal solution of the comparison differential system and also establishes sufficient conditions for the asymptotic practical stability of the trivial solution of a nonlinear impulsive Caputo fractional differential equations with fixed moments of impulse using the vector Lyapunov functions. First, it was discovered that the vector form of the Lyapunov function was majorized by the maximal solution of the comparison system. From the results obtained, it was also established that the main system is asymptotically practically stable in the sense of Lyapunov.

A Robust and Versatile Numerical Framework for Modeling Complex Fractional Phenomena: Applications to Riccati and Lorenz Systems

The fractional differential quadrature method (FDQM) with generalized Caputo derivatives is used in this paper to show a new numerical way to solve fractional Riccati equations and fractional Lorenz systems. Unlike previous FDQM applications that have primarily focused on linear problems, our work pioneers the use of this method for nonlinear fractional initial value problems. By combining Lagrange interpolation polynomials and discrete singular convolution (DSC) shape functions with the generalized Caputo operator, we effectively transform nonlinear fractional equations into algebraic systems. An iterative method is then utilized to address the nonlinearity. Our numerical results, obtained using MATLAB, demonstrate the exceptional accuracy and efficiency of this approach, with convergence rates reaching 10−8. Comparative analysis with existing methods highlights the superior performance of the DSC shape function in terms of accuracy, convergence speed, and reliability. Our results highlight the versatility of our approach in tackling a wider variety of intricate nonlinear fractional differential equations.

Martínez–Kaabar Fractal–Fractional Laplace Transformation with Applications to Integral Equations

This paper addresses the extension of Martinez–Kaabar (MK) fractal–fractional calculus (for simplicity, in this research work, it is referred to as MK calculus) to the field of integral transformations, with applications to some solutions to integral equations. A new notion of Laplace transformation, named MK Laplace transformation, is proposed, which incorporates the MK α,γ-integral operator into classical Laplace transformation. Laplace transformation is very applicable in mathematical physics problems, especially symmetrical problems in physics, which are frequently seen in quantum mechanics. Symmetrical systems and properties can be helpful in applications of Laplace transformations, which can help in providing an effective computational tool for solving such problems. The main properties and results of this transformation are discussed. In addition, the MK Laplace transformation method is constructed and applied to the non-integer-order first- and second-kind Volterra integral equations, which exhibit a fractal effect. Finally, the MK Abel integral equation’s solution is also investigated via this technique.

Fuzzy fractional shallow water wave equations: analysis, convergence of solutions, and comparative study with depth as triangular fuzzy number

Abstract We explore the integration of fuzzy fractional calculus into the modeling framework, recognizing its significance in capturing the inherent uncertainties and complexities present in Shallow Water Wave (ffSWW) dynamics. By incorporating fuzzy fractional calculus, we aim to enhance the accuracy and robustness of ffSWW equations, particularly in representing vague or imprecise parameters such as seabed topography, initial wave conditions, and material properties. In this article, we consider the time derivative as a fractional order instead of the traditional integer order, which allows us to interpret the behavior of the solution for different orders. Further, the sea depth has been considered as a Triangular Fuzzy Number (TFN). We employ the Homotopy Perturbation Transform Method (HPTM) to obtain the solution of ffSWW equations. The convergence of the obtained series solutions has been investigated theoretically and numerically. Also, the acquired results using the current method are validated through the comparison with pre-existing findings concerning integer order. Furthermore, simulation results for various fractional orders, as well as fuzzy lower and upper solutions of depth-averaged velocity and water surface elevation, are provided for triangular fuzzy numbers.

Existence and multiplicity for fractional differential equations with $$m(\xi )$$-Kirchhoff type-equation

Existence and multiplicity for fractional differential equations with $$m(\xi )$$-Kirchhoff type-equation

Analysis of shear creep damage model of slope rock mass under the influence of coupled damage effects (DBSM)

Abstract This article reveals the evolution law of shear creep damage of carbonaceous shale (weak interlayer of slope) under dynamic load from a mechanical perspective by combining indoor experiments and theoretical research. Firstly, a coupled damage variable D BSM was established for the initial damage D 0 and dynamic disturbance shear creep damage D BS of rock mass based on the theory of damage mechanics. Secondly, according to the fractional calculus operator theory and considering the influence of coupled damage variable D BSM on the viscosity coefficient of rock mass in the viscoelasticity and viscoplasticity creep stages, a shear creep damage model of rock mass was established. At the same time, combined with the shear creep test data under the influence of dynamic loads, the damage evolution law of shear creep in carbonaceous shale (weak interlayer of slope) was revealed, and the accuracy of the established shear creep damage model was verified. Finally, the damage evolution law of carbonaceous shale was quantitatively analyzed. Results show that: The shear creep damage model established in this article has unique advantages. The greater the initial damage, the more likely shear creep failure is to occur under the influence of dynamic disturbance and shear creep loads. Dynamic disturbance accelerates the damage of weak interlayers of slope. The cumulative coupling damage of carbonaceous shale (D ma ≤0.18) under dynamic disturbance and multi-stage shear creep loads exhibits an S-like evolution pattern. The shear creep damage mechanism of carbonaceous shale is characterized by obvious initial damage effect, dynamic disturbance damage effect, and stress response characteristics.

Asian Rainbow Option Pricing Problem Based on Uncertain Fractional Differential Equations

Option pricing is a core issue in option trading, which directly affects investors’ trading decisions and risk management. With the growing maturity of uncertainty theory, the research on option pricing problems based upon uncertainty theory has been gradually deepened. However, the volatility of stock price is not only related to the price at the current moment, but also related to all the previous stock price changes. Uncertain fractional differential equations can well portray the memory and history dependence of financial markets. Its nonlocal nature can more accurately describe the long-term memory effect and the phenomenon of spiky fat tails of stock market fluctuations. Thus, this paper mainly explores the pricing of Asian rainbow option in uncertain financial environments, where the underlying asset price is viewed as uncertain fractional processes. First, two models are presented for discussing the price of Asian rainbow option in two-asset cases, respectively. Then the pricing problems of different types of Asian rainbow options under [Formula: see text]-assets case are further discussed. Finally, empirical analysis is conducted on the proposed models with real stock data to verify the validity of the pricing formulas.

A shooting-Newton procedure for solving fractional terminal value problems

In this paper we consider the numerical solution of fractional terminal value problems: namely, terminal value problems for fractional differential equations. In particular, the proposed method uses a Newton-type iteration which is particularly efficient when coupled with a recently-introduced step-by-step procedure for solving fractional initial value problems, i.e., initial value problems for fractional differential equations. As a result, the method is able to produce spectrally accurate solutions of fractional terminal value problems. Some numerical tests are reported to make evidence of its effectiveness.

Numerical Technique Based on Operational Matrices of Fractional Integration Using ψ$$ \psi $$‐Shifted Chebyshev Polynomials

ABSTRACTIn this paper, we present a numerical scheme based on a modified form of shifted Chebyshev polynomials to find the numerical solution of a class of fractional differential equations. For this purpose, we work out operational matrices of fractional integration of ‐shifted Chebyshev polynomials obtained from shifted Chebyshev polynomials. Finally, the solution to the problem under consideration is obtained by solving a system of algebraic equations that results from the use of operational matrices of integration. The analysis of integer and non‐integer order differential equations is presented to show the convergence of the solution of fractional order differential equation to the corresponding solution of the integer order differential equation. At the end, we present some linear and non‐linear examples to validate the theoretical analysis. Non‐linear examples are solved using Quasilinearization and proposed numerical technique.

IoT based healthcare system using Fractional dung beetle optimization enabled deep learning for breast cancer classification

Breast cancer classification plays a crucial role in healthcare, especially in the diagnosis and monitoring of patients. Traditional methods for classifying breast cancer based on histopathological images often suffer from limited accuracy, which can hinder early detection and treatment. Hence, this paper devises a novel Internet of Things (IoT) based healthcare system using SqueezeNet_Fractional Dung Beetle Optimization (Squeeze_FDBO) for breast cancer detection. Initially, IoT network is simulated, and routing of the histopathological images to the Base Station (BS) is established utilizing FDBO, which is obtained by combining Dung Beetle Optimizer (DBO), and the Fractional Calculus (FC). At BS, breast cancer classification is done, where input is first processed by a bilateral filter. Then, blood cell segmentation is effectuated using LadderNet, and then, feature extraction is performed. Finally, the multigrade classification of breast cancer is executed utilizing SqueezeNet tuned by FDBO. The efficiency of Squeeze_FDBO is validated using various performance measures, and it is found to record an accuracy of 0.919, sensitivity of 0.913, specificity of 0.923, Negative Predictive Value (NPV) of 0.920, and Positive Predictive Value (PPV) of 0.908, and a better routing performance with energy of 0.405J, distance of 6.901m, and delay of 0.650mS.

Chaotic dynamics and optimal therapeutic strategies for Caputo fractional tumor immune model in combination therapy.

In this paper, a Caputo fractional tumor immune model of combination therapy is established. First, the stability and biological significance of each equilibrium point are analyzed, and it is demonstrated that chaos may arise under specific conditions. Combined with the mathematical definition of Caputo fractional differentiation (CFD), it is found that there is a high correlation between the chaotic phenomenon of the patient's condition and the sensitivity of the patient to the change in the state of the day. The bifurcation threshold of each parameter is determined through numerical simulation, and the Hopf bifurcation of direct competition coefficient and inhibition coefficient between tumor cells and host healthy cells is elaborated upon in detail. Subsequently, a novel method combining optimal control theory with the particle swarm optimization (PSO) algorithm is proposed for the optimal control of the tumor immune model in combination therapy. Finally, the Adams-Bashforth-Moulton (ABM) prediction correction method is utilized in numerical simulations which demonstrate that the introduction of the CFD alters the model dynamics. Furthermore, these results indicate that fractional calculus can effectively be applied to tumor immune models better to elucidate complex chaotic dynamics of tumor cell evolution. Concurrently, the PSO can be successfully integrated with optimal control theory to address optimization challenges in cancer treatment.

A parametrized approach to generalized fractional integral inequalities: Hermite–Hadamard and Maclaurin variants

This paper introduces a novel parametrized integral identity that forms the basis for deriving a comprehensive class of generalized fractional integral inequalities. Building on recent advancements in fractional calculus, particularly in conformable fractional integrals, our approach offers a unified framework for various known inequalities. The novelty of this work lies in its ability to generate new and more general inequalities, including Hermite–Hadamard-, Maclaurin-, and corrected Maclaurin-type inequalities, by selecting specific parameter values. These results extend the scope of fractional integral inequalities and provide new insights into their structure. To demonstrate the practical applicability and accuracy of the theoretical findings, we present a detailed numerical example along with graphical representations.

Presentation of the efficient scheme for solving fractional order telegraph problems

The Asymptotic Homotopy Perturbation Transform Method AHPTM is presented in this work. It is combined version of the Asymptotic Homotopy Perturbation Method AHPM and Laplace transformation. The focus of the work is the introduction of a new fast convergent scheme to obtain the solution of the fractional partial differential equations. Therefore, the first demonstration of the AHPTM is present for the solution of space-fractional telegraph equation (SFTE) in this work. The Caputo version of fractional derivatives has been utilized. Three test problems of the important fractional telegraph model were solved by this proposed scheme. The scheme of AHPTM worked without exploiting Ji. Huan He polynomials or Adomian polynomials. This application was elaborated by providing error estimates, a graphical presentation and tabulation of the results obtained by AHPTM. The comparison of results obtained by AHPTM with exact results is provided which indicated the accuracy of the scheme.

Study of the parametric effect of the wave profiles of the time-space fractional soliton neuron model equation arising in the topic of neuroscience

Time-space fractional nonlinear problems (T-SFNLPs) play a crucial role in the study of nonlinear wave propagation. Time-space nonlinearity is prevalent across various fields of applied science, nonlinear dynamics, mathematical physics, and engineering, including biosciences, neurosciences, plasma physics, geochemistry, and fluid mechanics. In this context, we examine the time-space fractional soliton neuron model (TSFSNM), which holds significant importance in neuroscience. This model explains how action potentials are initiated and propagated by axons, based on a thermodynamic theory of nerve pulse transmission. The signals passing through the cell membrane (CM) are proposed to take the form of solitary sound pulses, which can be represented as solitons. To investigate these soliton solutions, nonlinear fractional differential equations (NLFDEs) are transformed into corresponding partial differential equations (PDEs) using a fractional complex transform (FCT). The Kudryashov method is then applied to determine the wave profiles for the TSFSNM equation. We present 3D, 2D, contour, and density plots of the TSFSNM equation, and further analyze how fractional and time-space parameters influence these wave profiles through additional graphical representations. Kink, singular kink and different types of soliton solutions are successfully recovered through the Kudryashov method. The outcomes of various studies show that the applied method is highly efficient and well-suited for tackling problems in applied sciences and mathematical physics. Graphical representations, coupled with numerical data, reinforce the validity and accuracy of the technique. The proposed method is a convenient and powerful tool for handling the solution of nonlinear equations, making it particularly effective in exploring complex wave phenomena in diverse scientific fields.

A spline-based framework for solving the space–time fractional convection–diffusion problem

In this study we consider a spline-based collocation method to approximate the solution of fractional convection–diffusion equations which include fractional derivatives in both space and time. This kind of fractional differential equations are valuable for modeling various real-world phenomena across different scientific disciplines such as finance, physics, biology and engineering.The model includes the fractional derivatives of order between 0 and 1 in space and time, considered in the Caputo sense and the spatial fractional diffusion, represented by the Riesz-Caputo derivative (fractional order between 1 and 2). We propose and analyze a collocation method that employs a B-spline representation of the solution. This method exploits the symmetry properties of both the spline basis functions and the Riesz-Caputo operator, leading to an efficient approach for solving the fractional differential problem. We discuss the advantages of using Greville Abscissae as collocation points, and compare this choice with other possible distributions of points. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

Higher-order fractional equations and related time-changed pseudo-processes

We study Cauchy problems of fractional differential equations in both space and time variables by expressing the solution in terms of “stochastic composition” of the solutions to two simpler problems. These Cauchy sub-problems respectively concern the space and the time differential operator involved in the main equation. We provide some probabilistic and pseudo-probabilistic applications, where the solution can be interpreted as the pseudo-transition density of a time-changed pseudo-process. To extend our results to higher order time-fractional problems, we introduce stable pseudo-subordinators as well as their pseudo-inverses. Finally, we present our results in the case of more general differential operators and we interpret the results by means of a linear combination of pseudo-subordinators and their inverse processes.

A new self-organization of complex networks structure generalized by a new class of fractional differential equations generated by 3D-gamma function

In this note, we suggest a generalization of gamma function to be 3D-gamma function. As a consequence, some special functions are generalized including the Mittag-Leffler function. Moreover, we utilize the generalized Mittag-Leffler to extend the AB-fractional calculus. Examples are introduced to cover our theory. The solvability of abstract Riccati equation is considered, and Hyers-Ulam stability is discovered in the sequel. Based on the new study, we design new stable of Self-organization in complex networks (SOCNs).