Comparative Study of Fractional Heun’s Method and Fractional q-Heun’s Method for Solving Fractional Differential Equations

It is known that Caputo fractional differential equations play an important role in modeling many physical situation. The models represented by Caputo fractional differential equation in general are better and efficient models than its counterpart with integer derivative models. In this work, we consider nonlinear Caputo impulsive fractional differential equations with initial conditions. Further, the impulses occur in the nonhomogeneous term. Initially, we have computed the solution of the linear Caputo impulsive fractional differential equation explicitly using the method of mathematical induction. We have developed comparison results in terms of coupled lower and upper solutions when the nonlinear terms are sums of an increasing and decreasing functions of the unknown function. Finally, we have developed generalized monotone method for the Nonlinear Caputo Impulsive Fractional Differential Equations with initial conditions. This proves the existence coupled minimal and maximal solutions of the nonlinear problem. Finally, under uniqueness condition, we prove the existence of the unique solution of the nonlinear Caputo fractional im-pulsive differential equation with initial condition. Further, the interval of existence is guaranteed by the upper and lower solutions. In this work, we have obtained the basic tools to enable us to develop the generalized iterative method for the nonlinear Caputo fractional impulsive differential with initial conditions. The basic tools developed are the explicit solution of the corresponding linear Caputo fractional impulsive differential equations with initial condition. We have achieved this by applying Laplace transform method. Laplace transform method is the most suitable method since the Caputo fractional derivative is a convolution integral. This explicit form is useful in establishing the uniqueness of the solution of the linear Caputo fractional impulsive differential conditions. We have developed two comparison theorems which are useful in proving the monotonicity of the linear iterates that will arise in the generalized monotone method and the uniqueness of the solution of the nonlinear Caputo fractional impulsive differential equation. We have also presented some numerical results. See [8, 19] for results on generalized iterative method for Caputo fractional differential equations without impulses.

Fractional diffusion equations describe phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second-order diffusion equations. Fractional differential equations raise mathematical difficulties that have not been encountered in the analysis of second-order differential equations. There are two properties of fractional differential operators that make the analysis of fractional differential equations more complicated than that for second-order differential equations. These are (i) fractional differential operators are nonlocal operators, and (ii) the adjoint of a fractional differential operator is not the negative of itself. The wellposedness of a Galerkin weak formulation to fractional elliptic differential equations with a constant diffusivity coefficient and the error analysis for corresponding finite element methods were proved previously. Many subsequent works were carried out to extend the analysis to other numerical methods. A constant diffusivity coefficient has been assumed in all these works. In this paper we present a counterexample which shows that the Galerkin weak formulation loses coercivity in the context of variable-coefficient conservative fractional elliptic differential equations. Hence, the previous results cannot be extended to variable-coefficient conservative fractional elliptic differential equations. We adopt an alternative approach to prove the existence and uniqueness of the classical solution to the variable-coefficient conservative fractional elliptic differential equation and characterize the solution in terms of the classical solutions to second-order elliptic differential equations. Furthermore, we derive a Petrov--Galerkin weak formulation to the fractional elliptic differential equation. We prove that the bilinear form of the Petrov--Galerkin weak formulation is weakly coercive and so the weak formulation has a unique weak solution and is well posed. Finally, we outline potential application of these results in the development of numerical methods for variable-coefficient conservative fractional elliptic differential equations.

In recent years many physical phenomenons have been successfully modelled using fractional partial differential equations such as in electromagnetics and material sciences. Fractional partial differential equations involve derivative of noninteger order. Many researchers attempted to solve various types of fractional differential equations using various methods. In this paper, the conformable fractional derivative definition introduced by Khalil et.al [4] is used. The fractional derivative using this definition satisfies many properties that the usual derivative has. By applying some similar arguments on Fourier transform for solving partial differential equations, some modifications on the Fourier transform are constructed to handle the fractional order in a fractional differential equation. The modified Fourier transform which is developed satisfies similar properties with the Fourier transform in the classical sense. The application of the method to solve a fractional differential equation is given as an example.

Most physical systems in nature display inherently nonlinear and dynamical properties; hence, it would be difficult for nonlinear equations to be solved merely by analytical methods, which has given rise to the emerging of engrossing phenomena such as bifurcation and chaos. Conjointly, due to nonlinear systems’ exhibiting more exotic behavior than harmonic distortion, it becomes compelling to test, classify and interpret the results in an accurate way. For this reason, avoiding preconceived ideas of the way the system is likely to respond is of pivotal importance since this facet would have effect on the type of testing run and processing techniques used in nonlinear systems. Paradigms of nonlinear science may suggest that it is ‘the study of every single phenomenon’ due to its interdisciplinary nature, which is another challenge encountered and needs to be addressed by generating and designing a systematic mathematical framework where the complexity of natural phenomena hints the requirement of identifying their commonalties and classifying their various manifestations in different nonlinear systems. Studying such common properties, concepts or paradigms can enable one to gain insight into nonlinear problems, their essence and consequences in a broad range of disciplines all forthwith. Fractional differential equations associated with non-local phenomena in physics have arisen as a powerful mathematical tool within a multidisciplinary research framework. Fractional differential equations, as one extension of the fractional calculus theory, can yield the evolution of various systems properly, which reinforces its position in mathematics and science while setting stage for the description of dynamic, complicated and nonlinear events. Through the reflection of the systems’ actual properties, fractional calculus manifests unforeseeable and hidden variations, and thus, enables integration and differentiation, with the solutions to be approximated by numerical methods along with modeling and predicting the dynamics of multiphysics, multiscale and physical systems. Neural Networks (NNs), consisting of hidden layers with nonlinear functions that have vector inputs and outputs, are also considerably employed owing to their versatile and efficient characteristics in classification problems as well as their sophisticated neural network architectures, which make them capable of tackling complicated governing partial differential equation problems. Furthermore, partial differential equations are used to provide comprehensive and accurate models for many scientific phenomena owing to the advancements of data gathering and machine learning techniques which have raised opportunities for data-driven identification of governing equations derived from experimentally observed data. Given these considerations, while many problems are solvable and have been solved, efforts are still needed to be able to respond to the remaining open questions in the fields that have a broad range of spectrum ranging from mathematics, physics, biology, virology, epidemiology, chemistry, engineering, social sciences to applied sciences. With a view of different aspects of such questions, our special issue provides a collection of recent research focusing on the advances in the foundational theory, methodology and topical applications of fractals, fractional calculus, fractional differential equations, differential equations (PDEs, ODEs, to name some), delay differential equations (DDEs), chaos, bifurcation, stability, sensitivity, machine learning, quantum machine learning, and so forth in order to expound on advanced fractional calculus, differential equations and neural networks with detailed analyses, models, simulations, data-driven approaches as well as numerical computations.

On the fractional-order logistic equation

In this paper we describe a method to solve the linear non-homogeneous fractional differential equations (FDE), composed with Jumarie type Fractional Derivative, and describe this method developed by us, to find out Particular Integrals, for several types of forcing functions. The solutions are obtained in terms of Mittag-Leffler functions, fractional sine and cosine functions. We have used our earlier developed method of finding solution to homogeneous FDE composed via Jumarie fractional derivative, and extended this to non-homogeneous FDE. We have demonstrated these developed methods with few examples of FDE, and also applied in fractional damped forced differential equation. This method proposed by us is useful as it is having conjugation with the classical methods of solving non-homogeneous linear differential equations, and also useful in understanding physical systems described by FDE.

This book aims to establish a foundation for fractional derivatives and fractional differential equations. The theory of fractional derivatives enables considering any positive order of differentiation. The history of research in this field is very long, with its origins dating back to Leibniz. Since then, many great mathematicians, such as Abel, have made contributions that cover not only theoretical aspects but also physical applications of fractional calculus. The fractional partial differential equations govern phenomena depending both on spatial and time variables and require more subtle treatments. Moreover, fractional partial differential equations are highly demanded model equations for solving real-world problems such as the anomalous diffusion in heterogeneous media. The studies of fractional partial differential equations have continued to expand explosively. However we observe that available mathematical theory for fractional partial differential equations is not still complete. In particular, operator-theoretical approaches are indispensable for some generalized categories of solutions such as weak solutions, but feasible operator-theoretic foundations for wide applications are not available in monographs. To make this monograph more readable, we are restricting it to a few fundamental types of time-fractional partial differential equations, forgoing many other important and exciting topics such as stability for nonlinear problems. However, we believe that this book works well as an introduction to mathematical research in such vast fields.

We derive the general prolongation formula for system of fractional partial differential equations (FPDEs) with Caputo derivative involving m dependent variables and n independent variables. A systematic computational method to derive Lie point symmetries of nonlinear FPDEs with Caputo derivative is given, and we illustrate its applicability through system of time fractional diffusion equations and space fractional diffusion equation in Caputo sense. We observe that the set of Lie point symmetries for a given fractional differential equation in Caputo sense is same as the corresponding fractional differential equation in Riemann–Liouville sense. Also we construct their exact solutions in terms of Mittag–Leffler function by using the obtained Lie point symmetries.

This paper analyzes a class of fractional calculus of variations problems and their associated Euler-Lagrange (fractional differential) equations. Unlike the existing fractional calculus of variations which is based on the classical notion of fractional derivatives, the fractional calculus of variations considered in this paper is based on a newly developed notion of weak fractional derivatives and their associated fractional order Sobolev spaces. Since fractional derivatives are direction-dependent, using one-sided fractional derivatives and their combinations leads to new types of calculus of variations and fractional differential equations as well as nonstandard Neumann boundary operators. This paper establishes the well-posedness and regularities for a class of fractional calculus of variations problems and their Euler-Lagrange (fractional differential) equations. This is achieved first for one-sided Dirichlet energy functionals which lead to one-sided fractional Laplace equations, then for more general energy functionals which give rise to more general fractional differential equations.

Numerical solution of multi-order fractional differential equations with multiple delays via spectral collocation methods

The aim of this paper is to study the stability of fractional differential equations in Hyers-Ulam sense. Namely, if we replace a given fractional differential equation by a fractional differential inequality, we ask when the solutions of the fractional differential inequality are close to the solutions of the strict differential equation. In this paper, we investigate the Hyers-Ulam stability of two types of fractional linear differential equations with Caputo fractional derivatives. We prove that the two types of fractional linear differential equations are Hyers-Ulam stable by applying the Laplace transform method. Finally, an example is given to illustrate the theoretical results.

This article is presented for manifestation of the non-homogeneous linear fractional differential equation under fuzzy uncertainty. Taking the initial values and the coefficients of the fractional differential equations to be fuzzy numbers, the solutions are categorized into different problems and sub-problems. Based on the three different combinations of the fuzzy initial values and fuzzy coefficients, the theoretical foundation in this article is primarily divided into three major problems. Then, each problem again contains two different sub-problems according to the sign of the coefficients. Finally, the solutions for each of the sub-problems are given in the sense of two different cases of $${}^{\text{RL}}[\left(i\right)-\alpha ]$$ and $${}^{\text{RL}}[\left(ii\right)-\alpha ]$$ differentiability. The strong and the weak solution criteria have been discussed here. Also, the existence and uniqueness criterion for solution of the initial-valued fuzzy fractional differential equation has been established in this article. Finally, an EPQ model of deteriorated items is discussed for its production phase only as a proper validation of proposed theory. The greatness of the consideration of fuzzy fractional differential equation over crisp integer differential equation, fuzzy integer differential equation and crisp fractional differential equation to describe the EPQ model is established in this current article. In this context, a new defuzzification technique is developed to compare the fuzzy and crisp phenomena related to the EPQ model.

In order to explore the fractional differential equations in accounting informatization financial software, the author proposes a system for fractional diffusion wave equations and fractional differential equations, two numerical algorithms with higher precision are given, and the amount of computation is reduced at the same time. First, based on the equivalent integral form of the time fractional diffusion wave equation, using the fractional echelon method and the Crank-Nicolson method, for the time fractional diffusion wave equation, a finite difference scheme is designed, this format has second-order accuracy in both the temporal and spatial directions and is computationally stable. Numerical examples verify the accuracy and effectiveness of this format. Then when dealing with the initial value problem of fractional differential equations with Caputo derivative operator, convert it to the equivalent Voltera integral equation system, an initial approximate solution is obtained by a low-order method, derive the residual and error equations, the idea of applying the stepwise correction of spectral delay correction improves the numerical accuracy of the solution, at the same time, the Richard Askey integral equation is used to reduce the amount of calculation. At last, the high precision and effectiveness of the new method are verified by numerical experiments. Experiments show that: Starting from the equivalent integral form of the fractional diffusion wave equation, a second-order finite-difference scheme of the fractional-order diffusive wave equation is constructed, through numerical experiments, it is verified that the scheme has good accuracy and efficiency. In numerical solution, discrete integral equations have better numerical stability than differential equations, therefore, the format also has better stability. When taking different fractional derivative indices a=1.5 and a=1.8, it can be seen that the difference format constructed by the author, in the time direction, has second-order precision, as expected.

Vibrations of inhomogeneous anisotropic viscoelastic bodies described with fractional derivative models

In this paper, we consider a nonlinear fractional differential equation. This equation takes the form of the Bernoulli differential equation, where we use the Caputo fractional derivative of non-integer order instead of the first-order derivative. The paper proposes an exact solution for this equation, in which coefficients are power law functions. We also give conditions for the existence of the exact solution for this non-linear fractional differential equation. The exact solution of the fractional logistic differential equation with power law coefficients is also proposed as a special case of the proposed solution for the Bernoulli fractional differential equation. Some applications of the Bernoulli fractional differential equation to describe dynamic processes with power law memory in physics and economics are suggested.