On the convergence of two-dimensional fuzzy Volterra-Fredholm integral equations by using Picard method

In this paper we prove convergence of the method of successive approximations used to approximate the solution of nonlinear two-dimensional Volterra-Fredholm integral equations and define the notion of numerical stability of the algorithm with respect to the choice of the first iteration. Also we present an iterative procedure to solve such equations. Finally, the method is illustrated by solving some examples

Status of the differential transformation method

Further to a recent controversy on whether the differential transformation method (DTM) for solving a differential equation is purely and solely the traditional Taylor series method, it is emphasized that the DTM is currently used, often only, as a technique for (analytically) calculating the power series of the solution (in terms of the initial value parameters). Sometimes, a piecewise analytic continuation process is implemented either in a numerical routine (e.g., within a shooting method) or in a semi-analytical procedure (e.g., to solve a boundary value problem). Emphasized also is the fact that, at the time of its invention, the currently-used basic ingredients of the DTM (that transform a differential equation into a difference equation of same order that is iteratively solvable) were already known for a long time by the "traditional"-Taylor-method users (notably in the elaboration of software packages --numerical routines-- for automatically solving ordinary differential equations). At now, the defenders of the DTM still ignore the, though much better developed, studies of the "traditional"-Taylor-method users who, in turn, seem to ignore similarly the existence of the DTM. The DTM has been given an apparent strong formalization (set on the same footing as the Fourier, Laplace or Mellin transformations). Though often used trivially, it is easily attainable and easily adaptable to different kinds of differentiation procedures. That has made it very attractive. Hence applications to various problems of the Taylor method, and more generally of the power series method (including noninteger powers) has been sketched. It seems that its potential has not been exploited as it could be. After a discussion on the reasons of the "misunderstandings" which have caused the controversy, the preceding topics are concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages, references and further considerations adde

Some new results on nonlinear fractional iterative Volterra-Fredholm integro differential equations

In this paper, we establish some new results concerning the existence and uniqueness of the solutions of iterative nonlinear Volterra-Fredholm integro differential equations subject to the initial conditions. The fractional derivatives are considered in the Caputo sense. Also these new results are obtained by applying the GronwallBellman’s inequality and the Banach contraction fixed point theorem. Moreover, the results of references [16, 17, 27] appear as a special case of our results.Emerging Sources Citation Index (ESCI)MathScinetScopu

Solving Integral Equations by Means of Fixed Point Theory

The authors thank their respective universities. A.F. Roldan Lopez de Hierro is grateful to Ministerio de Ciencia e Innovacion by Project PID2020-119478GB-I00 and to Program FEDER Andalucia 2014-2020 by Project A-FQM-170-UGR20.One of the most interesting tasks in mathematics is, undoubtedly, to solve any kind of equations. Naturally, this problem has occupied the minds of mathematicians since the dawn of algebra. There are hundreds of techniques for solving many classes of equations, facing the problem of finding solutions and studying whether such solutions are unique or multiple. One of the recent methodologies that is having great success in this field of study is the fixed point theory. Its iterative procedures are applicable to a great variety of contexts in which other algorithms fail. In this paper, we study a very general class of integral equations by means of a novel family of contractions in the setting of metric spaces. The main advantage of this family is the fact that its general contractivity condition can be particularized in a wide range of ways, depending on many parameters. Furthermore, such a contractivity condition involves many distinct terms that can be either adding or multiplying between them. In addition to this, the main contractivity condition makes use of the self-composition of the operator, whose associated theorems used to be more general than the corresponding ones by only using such mapping. In this setting, we demonstrate some fixed point theorems that guarantee the existence and, in some cases, the uniqueness, of fixed points that can be interpreted as solutions of the mentioned integral equations.Instituto de Salud Carlos III Spanish Government European Commission PID2020-119478GB-I00Program FEDER Andalucia 2014-2020 A-FQM-170-UGR2

New Challenges Arising in Engineering Problems with Fractional and Integer Order

Mathematical models have been frequently studied in recent decades, in order to obtain the deeper properties of real-world problems. In particular, if these problems, such as finance, soliton theory and health problems, as well as problems arising in applied science and so on, affect humans from all over the world, studying such problems is inevitable. In this sense, the first step in understanding such problems is the mathematical forms. This comes from modeling events observed in various fields of science, such as physics, chemistry, mechanics, electricity, biology, economy, mathematical applications, and control theory. Moreover, research done involving fractional ordinary or partial differential equations and other relevant topics relating to integer order have attracted the attention of experts from all over the world. Various methods have been presented and developed to solve such models numerically and analytically. Extracted results are generally in the form of numerical solutions, analytical solutions, approximate solutions and periodic properties. With the help of newly developed computational systems, experts have investigated and modeled such problems. Moreover, their graphical simulations have also been presented in the literature. Their graphical simulations, such as 2D, 3D and contour figures, have also been investigated to obtain more and deeper properties of the real world problem

Stability analysis of new generalized mean-square stochastic fractional differential equations and their applications in technology

Stability theory has significant applications in technology, especially in control systems. On the other hand, the newly defined generalized mean-square stochastic fractional (GMSF) operators are particularly interesting in control theory and systems due to their various controllable parameters. Thus, the combined study of stability theory and GMSF operators becomes crucial. In this research work, we construct a new class of GMSF differential equations and provide a rigorous proof of the existence of their solutions. Furthermore, we investigate the stability of these solutions using the generalized Ulam-Hyers-Rassias stability criterion. Some examples are also provided to demonstrate the effectiveness of the proposed approach in solving fractional differential equations (FDEs) and evaluating their stability. The paper concludes by discussing potential applications of the proposed results in technology and outlining avenues for future research

Applied Mathematics and Fractional Calculus

In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the fractional calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as: fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. This is why the application of fractional calculus theory has become a focus of international academic research. This Special Issue "Applied Mathematics and Fractional Calculus" has published excellent research studies in the field of applied mathematics and fractional calculus, authored by many well-known mathematicians and scientists from diverse countries worldwide such as China, USA, Canada, Germany, Mexico, Spain, Poland, Portugal, Iran, Tunisia, South Africa, Albania, Thailand, Iraq, Egypt, Italy, India, Russia, Pakistan, Taiwan, Korea, Turkey, and Saudi Arabia

The 2nd International Conference on Mathematical Modelling in Applied Sciences, ICMMAS’19, Belgorod, Russia, August 20-24, 2019 : book of abstracts

The proposed Scientific Program of the conference is including plenary lectures, contributed oral talks, poster sessions and listeners. Five suggested special sessions / mini-symposium are also considered by the scientific committe

New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus

This reprint focuses on exploring new developments in both pure and applied mathematics as a result of fractional behaviour. It covers the range of ongoing activities in the context of fractional calculus by offering alternate viewpoints, workable solutions, new derivatives, and methods to solve real-world problems. It is impossible to deny that fractional behaviour exists in nature. Any phenomenon that has a pulse, rhythm, or pattern appears to be a fractal. The 17 papers that were published and are part of this volume provide credence to that claim. A variety of topics illustrate the use of fractional calculus in a range of disciplines and offer sufficient coverage to pique every reader's attention

Mathematical Methods, Modelling and Applications

This volume deals with novel high-quality research results of a wide class of mathematical models with applications in engineering, nature, and social sciences. Analytical and numeric, deterministic and uncertain dimensions are treated. Complex and multidisciplinary models are treated, including novel techniques of obtaining observation data and pattern recognition. Among the examples of treated problems, we encounter problems in engineering, social sciences, physics, biology, and health sciences. The novelty arises with respect to the mathematical treatment of the problem. Mathematical models are built, some of them under a deterministic approach, and other ones taking into account the uncertainty of the data, deriving random models. Several resulting mathematical representations of the models are shown as equations and systems of equations of different types: difference equations, ordinary differential equations, partial differential equations, integral equations, and algebraic equations. Across the chapters of the book, a wide class of approaches can be found to solve the displayed mathematical models, from analytical to numeric techniques, such as finite difference schemes, finite volume methods, iteration schemes, and numerical integration methods