Fractional calculus in analytical and numerical techniques Engineering essay
The need for fractional calculus in science and engineering fields has increased dramatically in recent decades. Complex systems with inherited properties and memory properties are best modeled using fractional calculus. This is due to the non-local nature of fractional order derivatives that is lacking in the whole order counterparts. Fractional differential equations have been an exciting area of applied mathematics. It provides very important tools for describing and studying natural phenomena, in the field of fractional calculus. authors are interested in the theory of fractional differential equations because they are abstract formulations for many problems in physics and hydrology. Dear colleagues, Non-linearity exists in all complex real-life phenomena. The construction and investigation of nonlinear mathematical models arising in physics, bioengineering, optics, fluid dynamics and other areas of science and engineering is necessary to understand the physical framework of the related real-life phenomena. The fractional calculus FC can be considered an old yet new topic. It is an old subject because, starting from some speculations of GW Leibniz 1695, 1697 and L. Euler 1730, it has been gradually developed until now. However, it can also be considered a new topic. The FC has only been the object of this since the 1970s. We introduce a new numerical method, based on Bernoulli polynomials, for solving multinomial fractional differential equations of variable order. The fractional derivative of variable order was considered in Caputo's sense, while the Riemann-Liouville integral operator was used to give approximations to the unknown function and the associated, 2.1-dimensional time fractional Bogoyavlensky-Konopelchenko BK equation. , which describes the interaction of wave propagation along the x-axis and y-axis. To obtain the exact solutions of the BK equation, we used a subequation method based on the Riccati equation, and for VO fractional differential equations of variable order, FDEs with a time t, space x or other variable-dependent order with successfully applied to investigate time and/or space-dependent dynamics. This study aims to provide an overview of the recent relevant literature and findings in the areas of primary definitions, models, numerical methods and. This article discusses the application of generalized polynomials for solving nonlinear systems of partial differential equations of fractional order with initial conditions. First, the solutions are expanded using generalized polynomials through an operational matrix. The Unknown Free Coefficients and Control Parameters of the A Feature Paper should be a substantial original paper covering several techniques or approaches. study general classes of fractional - calculus operators. Burcu and Arran Fernandez. 2024. Numerical and Analytical Methods for Differential Equations and Systems Fractal and, no. 1:59. https: The study of fraction calculus dates back to the time when Leibnitz and Newton invented differential calculus. Fractional calculus deals with derivatives and integrals of arbitrary real order. It is a powerful tool for modeling phenomena arising in various fields such as mechanics, physics, engineering, economics, finance, medicine. Fractional calculus and its applications are fascinating areas of research in manytechnical disciplines. This book is a comprehensive collection of research from the author's group, which is one of the most active in the fraction calculus community worldwide and is the birthplace of one of the four MATLAB toolboxes in fractions. The neutron diffusion equation NDE is one of the most important partial differential equations PDEs, to describe the neutron behavior in nuclear reactors and many physical phenomena. In this article we reformulate this problem via Caputo fractional derivative with integer order initial conditions, the physical meaning of which is in this case. This textbook highlights the theory of fractional calculus and its broad applications in mechanics and engineering. It describes in detail the research results in using fractional calculus methods for modeling and numerical simulation of complex mechanical behavior. It covers the mathematical basis of fractional calculus, the Fractional Calculus FC is a generalization of classical calculus that deals with operations of integration and differentiation of non-integer, fractional order. The concept of fractional. This article deals with the numerical solutions of a class of fractional mathematical models that originated in the engineering sciences and are governed by time-fractional advection-diffusion-reaction TF-ADR equations, involving the Caputo derivative. In particular, we are interested in the models that connect chemical and hydrodynamic. In this study, a numerical technique is proposed for solving the generalized time-fractional Navier-Stokes equation in cylindrical coordinates via cubic B-spline CBS basis functions. The proposed technique to solve the problem is derived from the finite difference method and the collocation method. A fractional calculus approach to the dynamics of self-similar proteins. Article. Walter G. Gl ckle. TF Nonnenmacher. Request PDF, fractional order systems, numerical techniques, and. Normally, fractional differential equations are difficult to solve analytically, and with variable order fractional derivatives they become more challenging. Therefore, the need for reliable numerical techniques is worth investigating. To solve these types of equations, we derive a new approach based on the operational matrix. Fractional Calculus and Fractional Differential Equations FDE has many applications in various branches of science. But often a true nonlinear FDE does not have the exact or analytical solution. In addition, we compare the numerical results obtained with this technique for different values of alpha, showing that as the value moves from fractional order to integer order, the solution becomes increasingly similar to the exact solution. Additionally, we provide a tabular view of the solution for each example. Resume. In this paper we propose the shifted Legendre orthonormal polynomials for the numerical solution of the fractional optimal control problems encountered in various branches of physics. Yokus et al. 30 investigated several methods, including analytical, numerical, and approximate analytical techniques, to solve the time-fractional nonlinear Burger–Fisher equation. VO fractional differential equations of variable order FDEs with time t, space x or other variable-dependent order have been successfully applied to investigate time- and/or space-dependent dynamics. This study aims to provide an overview of the recent relevant literature and findings in the areas of primary definitions, models, numerical methods and efficient analytical and numerical methods. is important in many areas of applied science and engineering today.