Differential Equations Of Fractional Order Research Articles

On the Solutions of Nonlinear Implicit ω-Caputo Fractional Order Ordinary Differential Equations with Two-Point Fractional Derivatives and Integral Boundary Conditions in Banach Algebra

This article delves into the analysis of nonlinear implicit ω-Caputo fractional-order ordinary differential equations (NLIFDEs) with two-point fractional derivatives and integral boundary conditions within the context of Banach algebra. The primary focus is on demonstrating the existence and uniqueness of solutions for these complex fractional differential equations by utilizing Banach's and Krasnoselskii's fixed point theorems. Furthermore, the study explores the stability of these solutions through the Ulam-Hyers and Ulam-Hyers-Rassias stability criteria, thereby assessing the robustness of the proposed model. To illustrate the versatility of the generalized model, several special cases are examined, showcasing its ability to encompass various classical models. The practical applicability of the theoretical findings is underscored through a numerical example, which demonstrates the feasibility and relevance of the proposed methodology. This thorough investigation advances the comprehension of nonlinear fractional differential equations with integral boundary conditions, highlighting the intricate relationship between fractional derivatives, nonlinearities, and integral terms. The results offer significant insights into the behavior and stability of solutions within this demanding mathematical framework.

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.

Advanced optimal control approaches for immune boosting and clinical treatment to enhance dengue viremia models using ABC fractional-order analysis.

This work focuses on the Dengue-viremia ABC (Atangana-Baleanu Caputo) fractional-order differential equations, accounting for both symptomatic and asymptomatic infected cases. Symptomatic cases are characterized by higher viremia levels, whereas asymptomatic cases exhibit lower viremia levels. The fractional-order model highlights memory effects and other advantages over traditional models, offering a more comprehensive representation of dengue dynamics. The total population is divided into four compartments: susceptible, asymptomatic infected, symptomatic infected, and recovered. The model incorporates an immune-boosting factor for asymptomatic infected individuals and clinical treatment for symptomatic cases. Positivity and boundedness of the model are validated, and both local and global stability analyses are performed. The novel Adams-Bash numerical scheme is utilized for simulations to rigorously assess the impact of optimal control interventions. The results demonstrate the effectiveness of the proposed control strategies. The reproduction numbers must be reduced based on specific optimal control conditions to effectively mitigate disease outbreaks. Numerical simulations confirm that the optimal control measures can significantly reduce the spread of the disease. This research advances the understanding of Dengue-viremia dynamics and provides valuable insights into the application of ABC fractional-order analysis. By incorporating immune-boosting and clinical treatment into the model, the study offers practical guidelines for implementing successful disease control strategies. The findings highlight the potential of using optimal control techniques in public health interventions to manage disease outbreaks more effectively.

Impact of fractional and integer order derivatives on the [formula omitted]-dimensional fractional Davey–Stewartson–Kadomtsev–Petviashvili equation

In this study, the closed-form wave solutions of the (4+1)-dimensional fractional Davey–Stewartson–Kadomtsev–Petviashvili equation are investigated using the modified auxiliary equation method and the Jacobi elliptic function method. In the analysis, two fractional derivatives known as M-truncated, beta and integer order derivative are used. The fractional-order partial differential equation is transformed into an integer-order ordinary differential equation by using the wave transformation, fractional derivatives, and integer-order derivatives. As a result, wave function solutions are found, including bell shape, W-shaped, composite dark-bright and periodic wave. The effects of free parameters on the amplitudes and wave behaviors are illustrated. It is demonstrated extensively that changes in the free parameters lead to changes in the wave amplitude. A comparison of solutions using the two types of fractional derivatives and the integer-order derivatives is included. The effects of the beta derivative, the M-truncated derivative and integer order derivative on the considered model are presented using 2D and 3D figures.

A study on the fractional-order COVID-19 SEIQR model and parameter analysis using homotopy perturbation method

In this article, we present a fractional-order susceptible-exposed-infected-quarantine-recovered (SEIQR) model to analyze the dynamics of the COVID-19 pandemic. The model includes susceptible (S), exposed (E), infected (I), quarantined (Q), and recovered (R) populations and uses a fractional-order differential equation to provide a further accurate representation of the disease's progression. We employ the homotopy perturbation method (HPM) to derive analytical solutions and the Runge-Kutta fourth-order (RK4) method to obtain numerical solutions. The results indicate that the fractional-order model, particularly for a fractional parameter α = 0.40, provides better accuracy and stability compared to the classical integer-order model. This study highlights the importance of fractional-order modeling in understanding the spread of COVID-19 and suggests its potential application in predicting and controlling future epidemics.

Algorithm 1047: FdeSolver, a Julia Package for Solving Fractional Differential Equations

We introduce FdeSolver, an open-source Julia package designed to solve fractional-order differential equations efficiently. The available solutions are based on product-integration rules, predictor–corrector algorithms, and the Newton-Raphson method. The package covers solutions for one-dimensional equations with orders of positive real numbers. For higher-dimensional systems, it supports orders up to one. Incommensurate derivatives are allowed and defined in the Caputo sense. Here, we summarize the implementation for a representative class of problems and compare it with available alternatives in Julia and MATLAB. Moreover, FdeSolver leverages the power and flexibility of the Julia environment to offer enhanced computational performance, and our development emphasizes adherence to the best practices of open research software. To highlight its practical utility, we demonstrate its capability in simulating microbial community dynamics and modeling the spread of COVID-19. This latter application involves fitting the order of derivatives grounded on real-world epidemiological data. Overall, these results highlight the efficiency, reliability, and practicality of the FdeSolver Julia package.

A Two-Dimensional Nonlocal Fractional Parabolic Initial Boundary Value Problem

In this paper, we investigate a two-dimensional singular fractional-order parabolic partial differential equation in the Caputo sense. The partial differential equation is supplemented with Dirichlet and weighted integral boundary conditions. By employing a functional analysis method based on operator theory techniques, we prove the existence and uniqueness of the solution to the posed nonlocal initial boundary value problem. More precisely, we establish an a priori bound for the solution from which we deduce the uniqueness of the solution. For proof of its existence, we use various density arguments.

Combining approach of collocation and finite difference methods for fractional parabolic PDEs

This research aims to estimate the solutions of fractional-order partial differential equations of spacial fractional and both time-space fractional order. For this, we use finite differences for time derivatives and the well-known collocation method for space derivatives with lower-order Bernstein polynomials as basis functions. We explain the mathematical formulations in detail. Convergence and stability analysis of the space–time fractional diffusion equation with the source term is reported subsequently. Three numerical examples are considered for demonstrating the accuracy and reliability of the proposed method.

Unique Solutions for Caputo Fractional Differential Equations with Several Delays Using Progressive Contractions

We take into account a nonlinear Caputo fractional-order differential equation including several variable delays. We examine whether the solutions to the Caputo fractional-order differential equation taken under consideration, which has numerous variable delays, are unique. In the present study, first, we will apply the method of progressive contractions, which belongs to T.A. Burton, to Caputo fractional-order differential equation, including multiple variable delays, which has not yet appeared in the relevant literature by this time. The significant point of the method of progressive contractions consists of a very flexible idea to discuss the uniqueness of solutions for various mathematical models. Lastly, we provide two examples to demonstrate how this paper’s primary outcome can be applied.

Investigation of Fractional Order Tumor Cell Concentration Equation Using Finite Difference Method

المعادلة التفاضلية الكسرية توفر إطارًا رياضيًا قويًا لنمذجة وفهم مجموعة واسعة من الظواهر المعقدة وغير المحلية والمعتمدة على الذاكرة في مختلف التطبيقات العلمية والهندسية والعالمية. تعد الاعتلالات الكبدية أورامًا سرطانية ثانوية تتطور في الكبد نتيجة انتشار خلايا السرطان منورام سرطاني أصلي ينشأ في جزء آخر من جسم الإنسان. التركيز الأساسي لهذه الدراسة هو فهم نمو الأورام في الكبد البشري، سواء بوجود أو بدون علاج دوائي. ولتحقيق ذلك، يتم استخدام معادلة تفاضلية جزئية معرفة بالترتيب الزمني والكسري، ويتم تحليلها باستخدام أساليب عددية. يتم استخدام المشتقة كابوتو لاستكشاف تأثير العلاج الدوائي على نمو الورم. لحل النموذج الرياضي عدديًا، تم تطوير طريقة الفروق المحدودة كرانك-نيكولسون. يتم اختيار هذه الطريقة بسبب خصائصها المميزة، بما في ذلك الاستقرار الغير مشروط والدقة من الدرجة الثانية في الأبعاد الزمنية والمكانية. الاستقرار والدقة التي توفرهما هاتان الخصائص أمران حاسمان لضمان موثوقية النتائج المحصلة في هذه البحث. يتم تقديم النتائج والتحققات المستمدة من هذه الدراسة بفعالية من خلال تمثيلات بصرية متنوعة. تعتبر هذه الوسائل البصرية أدوات لا غنى عنها لفهم التأثير العميق للعلاج الدوائي على نمو الأورام. من خلال هذا التمثيل البصري، يمكن للشخص الحصول على فهم أوضح وأكثر تفصيلًا للديناميات المعقدة التي تحدث. تم الوصول إلى الحل العددي لهذه المشكلة المعقدة من خلال تنفيذ خوارزميات تم تطويرها بدقة باستخدام لغة البرمجة بايثون المتعددة الاستخدامات والقوية. مرونة بايثون والمكتبات الشاملة وإمكانيات الحساب العددي القوية تجعلها الخيار المثالي للتعامل مع تعقيدات هذه الدراسة.

Analysis of Solutions for Nonlinear ψ-Caputo Fractional Differential Equations with Fractional Derivative Boundary Conditions in Banach Algebra

This article explores the solutions of nonlinear implicit ψ-Caputo fractional-order ordinary differential equations (NLIFDEs) with two-point fractional derivatives boundary conditions in Banach algebra. The research aims to establish the existence and uniqueness of solutions for this complex class of differential equations. Utilizing Banach’s and Krasnoselskii’s fixed point theorems, the study conducts a rigorous analysis of the solutions, ensuring their existence and uniqueness. This comprehensive investigation contributes to enhancing the understanding of the behavior of solutions of nonlinear fractional differentials within a challenging mathematical framework.

Simultaneous space–time Hermite wavelet method for time-fractional nonlinear weakly singular integro-partial differential equations

An innovative simultaneous space–time Hermite wavelet method has been developed to solve weakly singular fractional-order nonlinear integro-partial differential equations in one and two dimensions with a focus whose solutions are intermittent in both space and time. The proposed method is based on multi-dimensional Hermite wavelets and the quasilinearization technique. The simultaneous space–time approach does not fully exploit for time-fractional nonlinear weakly singular integro-partial differential equations. Subsequently, the convergence analysis is challenging when the solution depends on the entire time domain (including past and future time), and the governing equation is combined with Volterra and Fredholm integral operators. Considering these challenges, we use the quasilinearization technique to handle the nonlinearity of the problem and reconstruct it to a linear integro-partial differential equation with second-order accuracy. Then, we apply multi-dimensional Hermite wavelets as attractive candidates on the resulting linearized problems to effectively resolve the initial weak singularity at t=0. In addition, the collocation method is used to determine the tensor-based wavelet coefficients within the decomposition domain. We elaborate on constructing the proposed simultaneous space–time Hermite wavelet method and design comprehensive algorithms for their implementation. Specifically, we emphasize the convergence analysis in the framework of the L2 norm and indicate high accuracy dependent on the regularity of the solution. The stability of the proposed wavelet-based numerical approximation is also discussed in the context of fractional-order nonlinear integro-partial differential equations involving both Volterra and Fredholm operators with weakly singular kernels. The proposed method is compared with existing methods available in the literature. Specifically, we highlighted its high accuracy and compared it with a recently developed hybrid numerical approach and finite difference methods. The efficiency and accuracy of the proposed method are demonstrated by solving several highly intermittent time-fractional nonlinear weakly singular integro-partial differential equations.

Corrections to “Complex order fractional differential equation in complex domain with mixed boundary condition”

The purpose of this note is to address and correct some errors identified in the recent paper by Yadav et al. (2024). In general, the set of sufficient conditions provided for ensuring the existence and uniqueness of solutions may not be adequate. Additionally, an incorrect inequality was used in the proof to demonstrate that T is a contraction mapping on B. This mistake is then carried forward into the analysis of (generalized) Ulam–Hyers stability. We therefore revisit the problem, establish the existence and uniqueness of solutions using the Banach fixed point theorem, and analyze the Ulam–Hyers stability.

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.

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.

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.

Boundary problems of sequential fractional differential equations having a monomial coefficient

The article examines sequential fractional-order differential equations composed of monomial coefficients defined as nonlinear boundary problems. The original problem is converted into an integral equation in an equivalent form. The contraction mapping principle and Krasnoselskii's fixed-point theorem are implemented to obtain two existence conditions. The problem of Ulam-Hyers stability is also investigated. An illustration is presented to showcase practical application of the obtained results.

Vibration suppression of a platform by a fractional type electromagnetic damper and inerter-based nonlinear energy sink

The theory of linear and nonlinear dynamic vibration absorbers is a well-established topic for many years. However, many recent contributions paid attention to the nonlinear vibration absorbers and different practical realizations of corresponding devices. Here, we propose a mechanical system constituted of the inerter-based nonlinear energy sink attached to the main body that is resting on an elastic foundation and is grounded through the fractional type electromagnetic damper. The two-degree-of-freedom system is described via two coupled differential equations with one of them having a fractional-order derivative term and the other one containing cubic stiffness nonlinearity. The incremental harmonic balance (IHB) method is employed to solve the equations and studies the strongly nonlinear periodic responses of the system. Applied approximated solution methodology is validated by the numerical Newmark method adapted to deal with the system of nonlinear fractional-order differential equations. The appropriate and necessary number of harmonics used in the IHB solution is commented and validated. This study can be a first step in understanding the dynamics and giving directions for the future design of vibration-isolating platforms based on inerter-based nonlinear vibration absorbers and electromagnetic dampers.

On fractional Pennes bio-heat equation using Legendre collocation method

In this paper, the distribution of heat in skin tissue during thermal treatment using the time-fractional Pennes model is studied. Temperature variations concerning several parameters such as derivative [Formula: see text] time, blood perfusion, and transmitted power are described using the Legendre collocation technique. The best feature of the proposed technique is the conversion of fractional-order differential equations into a set of algebraic systems and the ability to approximate both spatial and temporal coordinates with a single basis. The obtained results and comparison with an exact solution are illustrated.

Analytical Solution of Fractional-Order Lane-Emden Type Pantograph Delay Differential Equation

Analytical Solution of Fractional-Order Lane-Emden Type Pantograph Delay Differential Equation