Non-integer Order Differential Equation Research Articles

On Applicability of the Relaxation Spectrum of Fractional Maxwell Model to Description of Unimodal Relaxation Spectra of Polymers.

The relaxation time and frequency spectra are vital for constitutive models and for insight into the viscoelastic properties of polymers, since, from the spectra, other material functions used to describe rheological properties of various polymers can be uniquely determined. In recent decades the non-integer order differential equations have attracted interest in the description of time-dependent processes concerning relaxation phenomena. The fractional Maxwell model (FMM) is probably the most known rheological model of non-integer order. However, the FMM spectrum has not yet been studied and used to describe rheological materials. Therefore, the goal of the present paper was to study the applicability of the relaxation spectrum of FMM to the description of the relaxation spectra of polymers. Based on the known integral representation of the Mittag-Leffler two-parameter function, analytical formulas describing relaxation time and frequency spectra of FMM model were derived. Monotonicity of the spectra was analyzed and asymptotic properties were established. Relaxation frequency spectrum grows for large frequencies with a positive power law, while the relaxation time spectrum decays for large times with a negative power of time. Necessary and sufficient conditions for the existence of the local extrema of the relaxation spectra were derived in the form of two trigonometric inequalities. A simple procedure for checking the existence or absence of the spectra extrema was developed. Direct analytical formulas for the local extrema, minima, and maxima are given in terms of model fractional and viscoelastic parameters. The fractional model parameters, non-integer orders of the stress and strain derivatives of FMM uniquely determine the existence of the spectrum extrema. However, the viscoelastic parameters of the FMM, elastic modulus, and relaxation time affect the maxima and minima of the relaxation spectra and the values of their local peaks. The influence of model parameters on their local extrema was examined. Next, the applicability of the continuous-discrete spectrum of FMM to describe Baumgaertel, Schausberger and Winter (BSW) and unimodal Gauss-like relaxation spectra, commonly used to describe rheological properties of various polymers, was examined. Numerical experiments have shown that by respective choice of the FMM parameters, in particular by respective choice of the orders of fractional derivatives of the stress and strain, a good fit for the relaxation modulus experiment data was obtained for polymers characterized both by BSW and Gauss-like relaxation spectra. As a result, a good approximation of the real spectra was reached. Thus, the viscoelastic relaxation spectrum of FMM, due to the availability of the two extra degrees of freedom (non-integer orders of the stress and strain derivatives), provides deep insights into the complex behavior of polymers and can be applied for a wide class of polymers with unimodal relaxation spectra.

A Comparative Study on Solution Methods for Fractional order Delay Differential Equations and its Applications

Fractional calculus is one of the evolving fields in applied sciences. Delay differential equation of non-integer order plays a vital role in epidemiology, population growth, physiology economy, medicine, chemistry, control, and electrodynamics and many mathematical modeling problems Fractional Delay differential equations usually lacks analytic solutions and some of these equations can only be solved by some numerical methods. In this review article we present a comparative study on some standard numerical methods applied to solve linear fractional order differential equations with time delay. Fractional finite difference method (FFDM), Predictor-corrector method (PCM) with new and extended versions has been discussed in this article. All above mentioned methods use the Caputo type fractional differential operator to define fractional derivatives. Solution of a real-life problem formulated by FDDEs has been discussed under these methods. Results have been presented in tabular and graphical form to analyze the efficiency and scarcity of mentioned methods. These graphical and numerical comparisons are provided to illustrate and corroborate the similarity and differences between these methods.

RLC electric circuit model of fractional order: a Green's function approach

This article uses a hybrid approach that combines Green's function and a fixed point scheme to obtain a semi-analytical solution to the RLC electrical circuit's non-integer order differential equation. The fractional differential equation's order is determined by the two parameters and . The proposed method is unique because it can be extended to general nonlinear fractional systems. Another highly desirable feature of the proposed approach is that the construction of Green's function depends on both endpoints of the time domain, and hence the produced approximate solutions are highly accurate for fractional boundary value problems. We provide examples with figures and tables to demonstrate the effectiveness of our method in solving electrical circuit problems. TThe obtained approximate solutions are compared with exact solutions or numerical simulations to validate the applicability and accuracy of the proposed scheme.

Solitary wave solutions for the originating waves that propagate of the fractional Wazwaz-Benjamin-Bona-Mahony system

In this article, we discuss various solitary wave solutions that are extremely important in applied mathematics. In order to construct accurate solution to nonlinear fractional PDEs, we have employed the Khater technique to nonlinear equation for 3-dimensional fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) using conformable fractional dervatives. We obtain novel sets of solutions by employing this method including both periodic kink and periodic singular kink solution, singular periodic waves solution, dark solitons, bell shaped soliton solution, bell shaped singular solution, bright solution and smooth periodic wave solutions. In order to comprehend the physical principles and significance of the technique, solutions have been graphically represented. The results acquired to demonstrate the efficiency of the computational technique for the WBBM equation, our findings unambiguously demonstrate that the suggested approach is a useful, potent and simple way for experimental result for various kinds of non-integer order differential equations in the engineering and applied sciences.

Convection heat–mass transfer of generalized Maxwell fluid with radiation effect, exponential heating, and chemical reaction using fractional Caputo–Fabrizio derivatives

Abstract This article is directed to analyze the transfer of mass and heat in a generalized Maxwell fluid flow unsteadily on a vertical flat plate oscillating in its respective plane and heated exponentially. It explains the transfer of mass and heat using a non-integer order derivative usually called a fractional derivative. It is a generalization of the classical derivatives of the famous Maxwell’s equation to fractional non-integer order derivatives used for one-dimensional flow of fluids. The definition given by Caputo–Fabrizio for the fractional derivative is used for solving the problem mathematically. The Laplace transform method is used for finding the exact analytical solution to a problem by applying it to a set of non-integer order differential equations that are dimensionless in nature. These equations contain concentration, temperature, and velocity equations with specific initial and boundary conditions. Solutions of the three equations are graphically represented to visualize the effects of various parameters, such as the radiation parameter (Nr), the thermal Grashof number, the fractional parameter (α), the mass Grashof number, Prandtl effective number, Schmidt number, Prandtl number, the chemical reaction ( η 2 ) ({\eta }_{2}) , mass, and the temperature during fluid flow.

Solid-Rotor Induction Motor Modeling Based on Circuit Model Utilizing Fractional-Order Derivatives

This paper presents the Park model of a solid-rotor induction motor. In this model, the dynamic state of the motor is described by integer and noninteger order differential equations. The skin effect in the solid rotor was represented by resistance and inductance with lumped constants, and the fractional inductance was dependent on the frequency of the eddy current induced in the rotor. The parameters of the equivalent circuit were determined by the standstill frequency response test with the stationary machine on the basis of the finite element method analysis of the electromagnetic field. A simulation of the dynamic states of the induction motor with a solid rotor was carried out based on the calculated parameters. The simulation was carried out using a program written in the Matlab environment. The simulations show that the electromagnetic moment during the motor start-up is about 2 times greater than the initial torque in the steady state. On the other hand, the maximum value of the stator current during the start-up is about 1.5 times greater than the effective value of the inrush current in the steady state. A good agreement was obtained between the results calculated from the distribution of the magnetic field by the finite element method and the results obtained on the basis of the equivalent circuit and, in the case of the electromagnetic torque, with the results obtained from the transient state during motor reversal.

Solution of fractional boundary value problems by $ \psi $-shifted operational matrices

<abstract><p>In this paper, a numerical method is presented to solve fractional boundary value problems. In fractional calculus, the modelling of natural phenomenons is best described by fractional differential equations. So, it is important to formulate efficient and accurate numerical techniques to solve fractional differential equations. In this article, first, we introduce $ \psi $-shifted Chebyshev polynomials then project these polynomials to formulate $ \psi $-shifted Chebyshev operational matrices. Finally, these operational matrices are used for the solution of fractional boundary value problems. The convergence is analysed. It is observed that solution of non-integer order differential equation converges to corresponding solution of integer order differential equation. Finally, the efficiency and applicability of method is tested by comparison of the method with some other existing methods.</p></abstract>

Existence results for coupled differential equations of non-integer order with Riemann-Liouville, Erdélyi-Kober integral conditions

<abstract><p>This paper proposes the existence and uniqueness of a solution for a coupled system that has fractional differential equations through Erdélyi-Kober and Riemann-Liouville fractional integral boundary conditions. The existence of the solution for the coupled system by adopting the Leray-Schauder alternative. The uniqueness of solution for the problem can be found using Banach fixed point theorem. In order to verify the proposed criterion, some numerical examples are also discussed.</p></abstract>

New Perspective on the Conventional Solutions of the Nonlinear Time-Fractional Partial Differential Equations

The role of integer and noninteger order partial differential equations (PDE) is essential in applied sciences and engineering. Exact solutions of these equations are sometimes difficult to find. Therefore, it takes time to develop some numerical techniques to find accurate numerical solutions of these types of differential equations. This work aims to present a novel approach termed as fractional iteration algorithm-I for finding the numerical solution of nonlinear noninteger order partial differential equations. The proposed approach is developed and tested on nonlinear fractional-order Fornberg–Whitham equation and employed without using any transformation, Adomian polynomials, small perturbation, discretization, or linearization. The fractional derivatives are taken in the Caputo sense. To assess the efficiency and precision of the suggested method, the tabulated numerical results are compared with the standard variational iteration method and the exact solution as well. In addition, numerical results for different cases of the fractional-order α are presented graphically, which show the effectiveness of the proposed procedure and revealed that the proposed scheme is very effective, suitable for fractional PDEs, and may be viewed as a generalization of the existing methods for solving integer and noninteger order differential equations.

A New Operational Matrices-Based Spectral Method for Multi-Order Fractional Problems

The operational matrices-based computational algorithms are the promising tools to tackle the problems of non-integer derivatives and gained a substantial devotion among the scientific community. Here, an accurate and efficient computational scheme based on another new type of polynomial with the help of collocation method (CM) is presented for different nonlinear multi-order fractional differentials (NMOFDEs) and Bagley–Torvik (BT) equations. The methods are proposed utilizing some new operational matrices of derivatives using Chelyshkov polynomials (CPs) through Caputo’s sense. Two different ways are adopted to construct the approximated (AOM) and exact (EOM) operational matrices of derivatives for integer and non-integer orders and used to propose an algorithm. The understudy problems have been transformed to an equivalent nonlinear algebraic equations system and solved by means of collocation method. The proposed computational method is authenticated through convergence and error-bound analysis. The exactness and effectiveness of said method are shown on some fractional order physical problems. The attained outcomes are endorsing that the recommended method is really accurate, reliable and efficient and could be used as suitable tool to attain the solutions for a variety of the non-integer order differential equations arising in applied sciences.

Study Boundary Problem with Integral condition for Fractional Differential Equations

In last many years ago there was a great interest in studying the existence of positive solutions for fractional differential equations. Many authors have considered the existence of positive solutions of non-linear differential equations of non-integer order with integral boundary value conditions using fixed point theorems. G.wang etal(2012)in vest gated the following fractional differential equation 〖^c〗D^α W(t)+f(t,W(t))=0,0 λ is a positive number (0 < λ < 2),〖^C〗D^αis the standard Caputo fractional derivative obtained his results by means of Guo-krosnosel'skii theorem in a cone also A.Cabada etat (2013) established the following non-linear fractional differential equation with integral boundary value conditions D^α W(t)+f(t,W(t))=0 ,00 ,λ≠α ,〗 D^αis Riemann –Liovuville standard fractional derivative and f is a continuous function the results was based on Guo-krasnosel'skii fixed point theorem in a cone . This paper we investigate the existence results of a positive solution for integral boundary value conditions of the following system of equations: 〖^c〗D^β h(t)+k(t,h(t))=0 ,t∈(0,1) h(0)=h^' (0)=h^''' (0)=0 ,h(1)=δ∫_0^1▒h(n)dn where 3< β≤4 ,δ is a positive number , δ≠3 ,〖^C〗D^β denotes Caputo standard derivative and k is a continuous function.Our work based on Banach's and Schauder's theorem.

A comparative study of the fractional oscillators

In this work, we have investigated the fractional differential equation to describe the motion of a linear oscillator using fractional derivative operators with or without singular kernels. In order to be consistent with the physical systems the value of the fractional parameter that characterizes the existence of fractional structures in the system, lies within unit interval. The solutions of the non-integer order differential equation are obtained and expressed in terms of generalized G function depending upon the fractional parameter. The classical cases could be recovered by making the limit of fractional parameter approaches to unity. Moreover, we will analyse and compare the behaviour of the oscillator with different definitions of the fractional operators via graphical illustrations, phase portraits and Poincare maps.

Application of topological degree method in quantitative behavior of fractional differential equations

In the present manuscript we incorporate fractional order Caputo derivative to study a class of non-integer order differential equation. For existence and uniqueness of solution some results from fixed point theory is on our disposal. The method used for exploring these existence results is topological degree method and some auxiliary conditions are developed for stability analysis. For further elaboration an illustrative example is provided in the last part of the research article.

On the initial value problem for fuzzy differential equations of non-integer order $$\alpha \in (1,2)$$

The purpose of this paper is to present some basic theories of an initial value problem of fuzzy fractional differential equations involving the Caputo-fuzzy-type concept of fractional derivative in the case of the order $$\alpha \in (1,2).$$ The existence and uniqueness results of the solution for the given problem are presented. Finally, some examples are given to illustrate our main results.

Comparison of fractional order techniques for measles dynamics

A mathematical model which is non-linear in nature with non-integer order ϕ, 0 < phi leq 1 is presented for exploring the SIRV model with the rate of vaccination mu _{1} and rate of treatment mu _{2} to describe a measles model. Both the disease free mathcal{F}_{0} and the endemic mathcal{F}^{*} points have been calculated. The stability has also been argued for using the theorem of stability of non-integer order differential equations. mathcal{R} _{0}, the basic reproduction number exhibits an imperative role in the stability of the model. The disease free equilibrium point mathcal{F}_{0} is an attractor when mathcal{R}_{0} < 1. For mathcal{R}_{0} > 1, mathcal{F}_{0} is unstable, the endemic equilibrium mathcal{F}^{*} subsists and it is an attractor. Numerical simulations of considerable model are also supported to study the behavior of the system.

Analysis and Optimal Control of Fractional-Order Transmission of a Respiratory Epidemic Model.

The World Health Organization is yet to realise the global aim of achieving future-free and eliminating the transmission of respiratory diseases such as H1N1, SARS and Ebola since the recent reemergence of Ebola in the Democratic Republic of Congo. In this paper, a Caputo fractional-order derivative is applied to a system of non-integer order differential equation to model the transmission dynamics of respiratory diseases. The nonnegative solutions of the system are obtained by using the Generalized Mean Value Theorem. The next generation matrix approach is used to obtain the basic reproduction number mathcal {R}_{0} . We discuss the stability of the disease-free equilibrium when mathcal {R}_{0} < 1, and the necessary conditions for the stability of the endemic equilibrium when mathcal {R}_{0} > 1 . A sensitivity analysis shows that mathcal {R}_{0} is most sensitive to the probability of the disease transmission rate. The results from the numerical simulations of optimal control strategies disclose that the utmost way of controlling or probably eradicating the transmission of respiratory diseases should be quarantining the exposed individuals, monitoring and treating infected people for a substantial period.

A Fundamental Approach on Non-integer Order Differential Equation Using Nonlocal Fractional Sub-Strips Boundary Conditions

The intention of this article is to exhibit the validity and applicability of a fractional sub-strips integral boundary condition on a nonlinear differential equation of fractional order. We propose convergent and stable methods to validate the existence and uniqueness of the stated problem. The scheme is also provided with an illustrative examples.

On the Use of Aboodh Transform for Solving Non-integer Order Dynamical Systems

In this research study, a transform technique called the Aboodh transform has been employed to find closed form solutions of initial value problems in non-integer order ordinary differential equations. A newly derived theorem in the present study provides a general structure for finding the exact solutions of such non-integer order problems to be solved by this transform technique. Upon comparison with existing transforms including Laplace, Mellin, Sumudu and Elzaki, the Aboodh transform can be found in good agreement. Various non-integer order differential equations including Bagley-Torvik type equation have successfully been solved by the Aboodh transform technique.

Research of the stability of some hereditary dynamic systems

In the training course of the theory of differential equations, there exists a section on the investigation of the stability of systems of differential equations. If the system of differential equations consists of differential equations of integer order, then the stability theory of Lyapunov is usually used to research the stability of their rest points. However, in the case when the system of differential equations consists of differential equations of non-integer order, then it is necessary to use other methods of investigating the stability of such systems. Therefore, this article is devoted to the method of investigating the rest points of systems of differential equations of fractional order. In this paper we will investigate the stability of the rest points of the hereditary dynamical systems by the example of some fractal oscillators. Moreover, we will consider two types of hereditary dynamical systems: commensurable and incommensurate, for which the corresponding stability theorems for rest points are valid. Next, examples of applying these stability theorems to a fractal linear oscillator, the Duffing fractal oscillator, will be considered. The results of the research of the stability of the rest points of the hereditary dynamical systems were confirmed by constructing phase trajectories for the fractal oscillators under consideration. This article can be useful in the study of a fairly new section in the theory of differential equations-fractional calculus.

An optimal method for approximating the delay differential equations of noninteger order

The main purpose of this paper is to use a method with a free parameter, named the optimal asymptotic homotopy method (OHAM), in order to obtain the solution of delay differential equations, delay partial differential equations, and a system of coupled delay equations featuring fractional derivative. This method is preferable to others since it has faster convergence toward homotopy perturbation method, and the convergence rate can be set as a controlled area. Various examples are given to better understand the use of this method. The approximate solutions are compared with exact solutions as well.