System Of Fractional Differential Equations Research Articles

Lyapunov-Type Inequalities for Systems of Riemann-Liouville Fractional Differential Equations with Multi-Point Coupled Boundary Conditions

We consider a system of Riemann–Liouville fractional differential equations with multi-point coupled boundary conditions. Using some techniques from matrix analysis and the properties of the integral operator defined on two Banach spaces, we establish some Lyapunov-type inequalities for the problem considered. Moreover, the comparison between two Lyapunov-type inequalities is given under certain special conditions. The inequalities obtained compliment the existing results in the literature.

Coupled Systems of Nonlinear Proportional Fractional Differential Equations of the Hilfer-Type with Multi-Point and Integro-Multi-Strip Boundary Conditions

In this paper, we study a coupled system of nonlinear proportional fractional differential equations of the Hilfer-type with a new kind of multi-point and integro-multi-strip boundary conditions. Results on the existence and uniqueness of the solutions are achieved by using Banach’s contraction principle, the Leray–Schauder alternative and the well-known fixed-point theorem of Krasnosel’skiĭ. Finally, the main results are illustrated by constructing numerical examples.

An Analysis of a Fractional-Order Model of Colorectal Cancer and the Chemo-Immunotherapeutic Treatments with Monoclonal Antibody

The growth of colorectal cancer tumors and their reactions to chemo-immunotherapeutic treatment with monoclonal antibodies (mAb) are discussed in this paper using a system of fractional order differential equations (FDEs). mAb medications are still at the research stage; however, this research takes into account the mAbs that are already in use. The major goal is to demonstrate the effectiveness of the mAb medication Cetuximab and the significance of IL-2 levels in immune system support. The created model is broken down into four sub-systems: cell populations, irinotecan (CPT11) concentration for treatment, IL-2 concentration for immune system support, and monoclonal antibody Cetuximab. We show the existence and uniqueness of the initial value problem (IVP). After that, we analyze the stability of the equilibrium points (disease-free and co-existing) using the Routh–Hurwitz criteria. In addition, in applying the discretization process, we demonstrate the global stability of the constructed system around the equilibrium points based on specific conditions. In the end, simulation results were carried out to support the theory of the manuscript.

Existence and H-U stability of a tripled system of sequential fractional differential equations with multipoint boundary conditions

In this paper, we introduce a new coupled system of sequential fractional differential equations with coupled boundary conditions. We establish existence and uniqueness results using the Leray–Schauder alternative and Banach contraction principle. We examine the stability of the solutions involved in the Hyers–Ulam type. As an application, we present a few examples to illustrate the main results.

An efficient numerical method based on Fibonacci polynomials to solve fractional differential equations

The Fibonacci sequence is significant because of the so-called golden ratio, which describes predictable patterns for everything. Fibonacci polynomials are related to Fibonacci numbers, and in this paper we extend their applicability by using them to solve fractional differential equations (FDEs) and systems of fractional differential equations (SFDEs). With the help of the Riemann–Liouville fractional integral operator for the fractional-order hybrid function of block-pulse functions and the Fibonacci polynomials defined in this paper, the solution of the considered FDE and SFDE is reduced to a system of algebraic equations, which can be solved by Newton’s iterative method. The fractional order is obtained by transforming x into xα, with α>0. Compared to other models, our method in some situations is better by twelve orders of magnitude. There are situations when we get the exact solution. The presented method proves to be simple and effective in solving nonlinear problems with given initial values.

Orbital b-metric spaces and related fixed point results on advanced Nashine–Wardowski–Feng–Liu type contractions with applications

In this article, we prove some novel fixed-point results for a pair of multivalued dominated mappings obeying a new generalized Nashine–Wardowski–Feng–Liu-type contraction for orbitally lower semi-continuous functions in a complete orbital b-metric space. Furthermore, some new fixed-point theorems for dominated multivalued mappings are established in the scenario of ordered complete orbital b-metric spaces. Some examples are offered to demonstrate the validity of our new results’ premise. To demonstrate the applicability of our findings, applications for a system of nonlinear Volterra-type integral equations and fractional differential equations are shown. These results extend the theoretical results of Nashine et al. (Nonlinear Anal., Model. Control 26(3):522–533, 2021).

The Analytic Solutions of the Fractional-Order Model for the Spatial Epidemiology of the COVID-19 Infection

This paper provides a mathematical fractional-order model that accounts for the mindset of patients in the transmission of COVID-19 disease, the continuous inflow of foreigners into the country, immunization of population subjects, and temporary loss of immunity by recovered individuals. The analytic solutions, which are given as series solutions, are derived using the fractional power series method (FPSM) and the residual power series method (RPSM). In comparison, the series solution for the number of susceptible members, using the FPSM, is proportional to the series solution, using the RPSM for the first two terms, with a proportional constant of ψ Γ n α + 1 , where ψ is the natural birth rate of the baby into the susceptible population, Γ is the gamma function, n is the n th term of the series, and α is the fractional order as the initial number of susceptible individuals approaches the population size of Ghana. However, the variation in the two series solutions of the number of members who are susceptible to the COVID-19 disease begins at the third term and continues through the remaining terms. This is brought on by the nonlinear function present in the equation for the susceptible subgroup. The similar finding is made in the series solution of the number of exposed individuals. The series solutions for the number of deviant people, the number of nondeviant people, the number of people quarantined, and the number of people recovered using the FPSM are unquestionably almost identical to the series solutions for same subgroups using the RPSM, with the exception that these series solutions have initial conditions of the subgroup of the population size. It is observed that, in this paper, the series solutions of the nonlinear system of fractional partial differential equations (PDEs) provided by the RPSM are more in line with the field data than the series solutions provided by the FPSM.

System of fractional boundary value problems at resonance

In this paper, we investigate the existence of solutions for a system of fractional differential equations at resonance set on the interval [0, 1]. We associate a Dirichlet condition at $$t=0$$ and an integral boundary condition at $$t=1$$ . Our main existence theorem relies on the coincidence degree theory. A detailed study of the linear operators involved in the fixed point formulation is carried out. Finally, an example is included to illustrate the applicability of theoretical result.

Comparison principles for systems of Caputo fractional order ordinary differential equations

The comparison principle for systems of Caputo fractional order ordinary differential equations still remains blank. By applying the Caputo solution continuation and Vainikko’s definition, and reconstructing the framework of system comparison, here we state and prove general comparison principles for systems of Caputo fractional order ordinary differential equations, and illustrate them by examples with numerical simulations.

On System of Variable Order Nonlinear p-Laplacian Fractional Differential Equations with Biological Application

The study of variable order differential equations is important in science and engineering for a better representation and analysis of dynamical problems. In the literature, there are several fractional order operators involving variable orders. In this article, we construct a nonlinear variable order fractional differential system with a p-Laplacian operator. The presumed problem is a general class of the nonlinear equations of variable orders in the ABC sense of derivatives in combination with Caputo’s fractional derivative. We investigate the existence of solutions and the Hyers–Ulam stability of the considered equation. The presumed problem is a hybrid in nature and has a lot of applications. We have given its particular example as a waterborne disease model of variable order which is analysed for the numerical computations for different variable orders. The results obtained for the variable orders have an advantage over the constant orders in that the variable order simulations present the fluctuation of the real dynamics throughout our observations of the simulations.

Representation of the solution of a system of fractional linear differential equations with Caputo derivatives for modeling processes in fractal media

An increasing amount of attention is currently being paid to the fundamentals of mathematical modeling of nonlocal heat and mass transfer processes in fractal media. Such physical processes are described by a system of fractional linear differential equations with Caputo derivatives. In this article we consider a linear Cauchy problem described by differential equations with fractional Caputo derivatives. Using an analogue of the Cauchy matrix we find an explicit representation of the solution to the problem in question.

A Novel Scheme of the ARA Transform for Solving Systems of Partial Fractional Differential Equations

In this article, a new analytical scheme of the ARA transform is introduced to solve systems of fractional partial differential equations. The principle of the proposed technique is based on combining the ARA transform with the residual power series method to create an approximate series solution for a system of partial differential equations of fractional order on the form of a rapid convergent series. To illustrate the effectiveness, accuracy, and validity of the suggested technique, an Attractive physical system, the fractional neutron diffusion equation with one delayed neutrons group, is discussed and solved. Two different neutron flux initial conditions are presented numerically to clarify various cases in order to ensure the theoretical results. The necessary Mathematica codes are run using vital nuclear reactor cross-section data, and the results for various values of time are tabulated and graphically represented.

On new aspects of Chebyshev polynomials for space-time fractional diffusion process

Abstract Chebyshev collocation scheme and Finite difference method plays central roles for solving fractional differential equations (FDE). Therefore purpose of this paper is to solve fractional mathematical problem of diffusion by Chebyshev collocation method which turns the original problems into the system of fractional ordinary differential and algebraic equations by imposing orthogonality property. This system is solved by implementing Finite difference method. The numerical illustrations confirm that the combination of these two methods allow us to establish one of the best truncated solution in the series form.

Fractional order model to study the impact of planting genetically modified trees on the regulation of atmospheric carbon dioxide with analysis and modeling

The hazard of global warming is principally caused by a rise in the atmospheric concentration of carbon dioxide (CO2). One of the main carbon dioxide sinks is forests. In this study, we examine how atmospheric carbon dioxide can be controlled as a result of genetically modified planting trees. We create a system of fractional differential equations that represents a time-fractional order model. Both qualitative and quantitative analyses are performed for the fractional order system. positiveness and boundedness of solutions of the fractional order model are verified. Fixed point theory is used to derive the uniqueness and existence of the fractional order model. Examination of Ulam–Hyers stability as well as the iterative scheme is performed by using fixed point theory. To study the effect of the fractional operator, which demonstrates the dynamical behavior of the model, solutions are generated. To demonstrate how important parameters affect the dynamics of the forest cover and atmospheric CO2 gas, numerical simulations have been created. Fractional order memory effect is examined will assist in creating conditions depending on the model’s parameters.

Implementation of non-linear mixed effects models defined by fractional differential equations.

Fractional differential equations (FDEs), i.e. differential equations with derivatives of non-integer order, can describe certain experimental datasets more accurately than classic models and have found application in pharmacokinetics (PKs), but wider applicability has been hindered by the lack of appropriate software. In the present work an extension of NONMEM software is introduced, as a FORTRAN subroutine, that allows the definition of nonlinear mixed effects (NLME) models with FDEs. The new subroutine can handle arbitrary user defined linear and nonlinear models with multiple equations, and multiple doses and can be integrated in NONMEM workflows seamlessly, working well with third party packages. The performance of the subroutine in parameter estimation exercises, with simple linear and nonlinear (Michaelis-Menten) fractional PK models has been evaluated by simulations and an application to a real clinical dataset of diazepam is presented. In the simulation study, model parameters were estimated for each of 100 simulated datasets for the two models. The relative mean bias (RMB) and relative root mean square error (RRMSE) were calculated in order to assess the bias and precision of the methodology. In all cases both RMB and RRMSE were below 20% showing high accuracy and precision for the estimates. For the diazepam application the fractional model that best described the drug kinetics was a one-compartment linear model which had similar performance, according to diagnostic plots and Visual Predictive Check, to a three-compartment classic model, but including four less parameters than the latter. To the best of our knowledge, it is the first attempt to use FDE systems in an NLME framework, so the approach could be of interest to other disciplines apart from PKs.

Euler Wavelet Method as a Numerical Approach for the Solution of Nonlinear Systems of Fractional Differential Equations

In this paper, a numerical approach for solving systems of nonlinear fractional differential equations (FDEs) is presented Using the Euler wavelets technique and associated operational matrices for fractional integration, we try to solve those systems of FDEs. The method’s major objective is to transform the nonlinear FDE into a nonlinear system of algebraic equations that is straightforward to solve with matrix techniques. The Euler wavelets are constructed using Euler polynomials, which have fewer terms than most other polynomials used to construct other types of wavelets, therefore, using Euler wavelets for the numerical approach provides sparse operational matrices. Thanks to the sparsity of those operational matrices, the proposed numerical approach requires less computation and takes less time to evaluate. The approach described here is also applicable to systems of fractional differential equations with variable orders. To illustrate the strength and performance of the method, four numerical examples are provided.

Systems of Sequential ψ1-Hilfer and ψ2-Caputo Fractional Differential Equations with Fractional Integro-Differential Nonlocal Boundary Conditions

In this paper, we introduce and study a new class of coupled and uncoupled systems, consisting of mixed-type ψ1-Hilfer and ψ2-Caputo fractional differential equations supplemented with asymmetric and symmetric integro-differential nonlocal boundary conditions (systems (2) and (13), respectively). As far as we know, this combination of ψ1-Hilfer and ψ2-Caputo fractional derivatives in coupled systems is new in the literature. The uniqueness result is achieved via the Banach contraction mapping principle, while the existence result is established by applying the Leray–Schauder alternative. Numerical examples illustrating the obtained results are also presented.

Two effective methods for solving nonlinear coupled time-fractional Schrödinger equations

The objective of this work is to implement two efficient techniques, namely, the Laplace Adomian decomposition method (LADM) and the modified generalized Mittag–Leffler function method (MGMLFM) on a system of nonlinear fractional partial differential equations (NFPDEs) to get an analytic-approximate solution. The nonlinear time-fractional Schrödinger equation (TFSE) and coupled fractional order Schrödinger-Korteweg-de Vries (Sch-KdV) equation are found in various areas such as quantum mechanics and physics. These equations describe different types of wave propagation like dust-acoustic waves, Langmuir and electromagnetic waves in plasma physics. Using the proposed methods, a convenient solution is established for the considered nonlinear fractional order models. The obtained analytic-approximate travelling-waves solutions and the effect of the fractional order α on the behaviour of these projected solutions are presented in some figures and tables along with the exact solution. We compare the approximate values with their corresponding values of the known exact solution and compute the absolute error. Consequently, we can deduce that the used methods are very efficient, reliable and simple to construct a series form that rapidly convergent to the exact solution, which indicates the advantages of the methods.

Dynamics of bioconvection radiative MHD flow with gyrotactic microorganisms: an intelligent computation approach

Bioconvection flows relate significantly to real-life problems, such as designing bio-conjugates, bio-microsystems, and bio-cells. This study investigates a mathematical characterization of the electrically time-dependent conducting fluid flow containing gyrotactic microorganisms with activation energy, mass, and heat transfer. It is assumed that the flow is towards the extended surface and affects thermal radiations. In model formulation, the Navier–Stokes equations are reduced to a system of ordinary differential equations (ODEs) using a similarity transformation approach. For reference solutions, the RK4 technique is used with Dorman Prince coefficients. We have designed a new backpropagation neural network architecture to formulate new solution models with Bayesian regularization. The effects of variations in two parameters, Schmidt number (sc) and Prandtl number (pr), are investigated. Our experimental outcomes are linked to the reference outputs. It was observed that there is a strong agreement between our solutions and the reference solutions. Our approach can be used for systems of fractional differential equations.

On a System of Sequential Caputo Fractional Differential Equations with Nonlocal Boundary Conditions

We obtain existence and uniqueness results for the solutions of a system of Caputo fractional differential equations which contain sequential derivatives, integral terms, and two positive parameters, supplemented with general coupled Riemann–Stieltjes integral boundary conditions. The proofs of our results are based on the Banach fixed point theorem and the Leray–Schauder alternative.