![]() It has the advantage of simulating physical phenomena that involve memory effects or non-local interactions. The Liouville–Caputo fractional operator is a generalization of the classical derivative operator and can be used to solve FDEs with non-integer order derivatives. The Liouville–Caputo fractional operator is considered a powerful tool for solving fractional differential equations (FDEs) that have been used widely for simulating different complex problems. There is a close relationship between these two definitions since they can be converted through some regularity assumption. Each of these definitions adheres to some advantages and disadvantageous over the other and the most widely used of these applications is the Liouville–Caputo and Riemann–Liouville operators. There are various definitions of the fractional order including the Riemann–Liouville operator, Grünwald–Letnikov operator, Liouville–Caputo operator and Weyl–Riesz operator. They can also be used to describe chaotic systems in physics and other fields. Additionally, other applications of fractional calculus in several branches of science and engineering include the simulation of the model of viscoelastic materials in engineering applications and financial markets in economics. employed an exponential Euler scheme for simulating the multi-delay Caputo–Fabrizio fractional-order differential equations with application in control theory. investigated the possible application of fractional definitions to simulate a class of nonlinear fractional Langevin equations with important application in fluid dynamics. In addition, fractional derivatives are useful in many fields such as physics, engineering, economics, and finance. This allows for more complex phenomena such as memory effects, diffusion processes, and chaotic systems that might be difficult to solve using traditional definitions. This means that instead of taking derivatives or integrals concerning a single variable (as in traditional calculus), fractional calculus allows for derivatives or integrals to be taken concerning multiple variables simultaneously. These definitions vary depending on the context in which it is used, which in general have a common definition as the study of derivatives and integrals with non-integer orders. However, over time, its usefulness has been recognized and there have been various definitions and properties. The concept of fractional calculus was initially met with skepticism due to its unfamiliarity and lack of intuitive understanding. It has been around since the late 17th century when Gottfried Leibniz first proposed the concept of fractional derivatives, which has developed into a powerful tool for simulating different physical problems in many areas such as physics, chemistry, engineering, economics, and biology. The technique proves to be a valuable approach that can be extended in the future for other fractional models having real applications such as the fractional partial differential equations and fractional integro-differential equations.įractional calculus is a branch of mathematics that deals with the study of derivatives and integrals of non-integer order. The method is shown to give better results than the other methods using a lower number of bases and with less spent time, and helped in highlighting some of the important features of the model. The outcomes of the numerical examples support the theoretical results and show the accuracy and applicability of the presented approach. In addition, the computational time is computed and tabulated to ensure the efficacy and robustness of the method. To test the effectiveness of the proposed technique, several examples are simulated using the presented technique and these results are compared with other techniques from the literature. An error analysis for the method is proved to verify the convergence of the acquired solutions. ![]() We first present the differentiation matrix of fractional order that is used to convert the problem and its conditions into an algebraic system of equations with unknown coefficients, which are then used to find the solutions to the proposed models. We adapt a collocation technique involving a new operational matrix that utilizes the Liouville–Caputo operator of differentiation and Morgan–Voyce polynomials, in combination with the Tau spectral method. The model is considered one of the important models to simulate the coupled oscillator and various other applications in science and engineering. In this research, we present a new computational technique for solving some physics problems involving fractional-order differential equations including the famous Bagley–Torvik method. ![]()
0 Comments
Leave a Reply. |