Grunwald Letnikov Formula Research Articles

A numerical scheme for time-fractional Allen–Cahn equation with application in phase separation

The present study proposes a numerical method to solve the time-fractional Allen–Cahn equation. The Grunwald–Letnikov formula is employed for discretizing the time-fractional derivative, whereas the finite difference scheme is used for spatial approximation. Theoretical analysis indicates that the proposed method is stable in the discrete L 2 -norm and the numerical results satisfy the energy dissipation property. Numerical simulations are conducted for validation of the proposed method. It has been observed that the solution profile approaches equilibrium for various fractional-order values in the range of ( 0 , 1 ) . Moreover, the fractional order values have a significant effect on the solution stabilization rate which is faster for larger values and slower for the smaller values.

Efficient difference method for time-space fractional diffusion equation with Robin fractional derivative boundary condition

In this paper, a numerical method is proposed to solve the time-space fractional diffusion equation with Robin fractional derivative boundary condition. Under the weak regularity assumptions of solution, we present a numerical scheme based on the L1 method for time discretization on graded mesh and the Grunwald-Letnikov formula for spatial discretization on uniform mesh. And a fast scheme for the considered problem is constructed based on the exponential summation approximation of the kernel function t−α. Meanwhile, a detailed analysis of stability and convergence is given. Then, the extrapolation method is applied to the space direction to make it reach the second-order accuracy. Finally, numerical experiments show that the proposed method is effective.

Дифференциальные игры дробного порядка с распределенными параметрами

Изучена задача преследования в дифференциальных играх дробного порядка с распределенными параметрами. Частные дробные производные по времени и пространственные переменные рассматриваются в смысле Римана-Лиувилля, при аппроксимации применяется формула Грюн-вальда-Летникова. Рассматривается задача попадания в некоторой положительной окраине терминального множества. Для решения применяется способ конечных различий. Аппроксимируются дробные производные Римана-Лиувилля по пространственным переменным на отрезке с помощью формулы Грюнвальда-Летникова. По достаточному признаку существования дробной производной получена разностная аппроксимация производной дробного порядка по времени. Аппроксимируя дифференциальную игру на явную разностную, получаем дискретную игру. Сформулирована соответствующая задача преследования для дискретной игры, полученная с помощью аппроксимации непрерывной игры. Определено понятие возможности завершения преследования, дискретной игры в смысле точного поимки. Получены достаточные условия для завершения преследования. При этом показано, что порядок аппроксимации по времени равен единице, а по пространственным переменным — двум. Доказано, если в дискретной игре по заданному начальному положению возможно завершение преследования в смысле точного поимки, то в непрерывной игре из соответствующего начального положения возможно завершение преследования. Предложена структура построения управлений преследования, которая обеспечит завершение игры за конечное время. Методы, применяемые для этой задачи, могут быть использованы для изучения дифференциальных игр, описываемых более общими уравнениями дробного порядка.

Finite volume method for mixed convection boundary layer flow of viscoelastic fluid with spatial fractional derivatives over a flat plate

A new fractional finite volume method is developed for the mixed convection boundary layer flow and heat transfer of viscoelastic fluid over a flat plate. The spatial fractional derivative of the Riemann–Liouville type is employed in the constitutive relation and modified Fourier’s law respectively. Nonlinear and coupled boundary layer governing equations are formulated with non-uniform boundary conditions. The discretized scheme combined with the shifted Grunwald–Letnikov formula is proved to be conditionally stable, further the convergence and accuracy of the numerical solutions are presented. Results demonstrate that space fractional derivative parameters have strong effects on the velocity and temperature distributions. Moreover, the viscoelastic fluid with spatial fractional derivative performs stress relaxation with distance from the intersections of velocity profiles.

A New Solution for Optimal Control of Fractional Convection–Reaction–Diffusion Equation Using Rational Barycentric Interpolation

This paper solves an optimal control problem governed by a fractional convection–reaction–diffusion partial differential equation. Using Lagrangian multipliers, necessary conditions are obtained, and then, Barycentric collocation method are applied for discretizing classical derivatives and Grunwald–Letnikov formula for fractional derivative. Barycentric interpolation is a class of Lagrange polynomial interpolation that is fast and deserves to be known as a method of polynomial interpolation and Grunwald–Letnikov formula is a basic extension of the derivative in fractional calculus. Numerical examples are presented to show the effectiveness of the method.

Numerical investigation of the nonlinear modified anomalous diffusion process

The nonlinear modified anomalous sub-diffusion model characterizes processes that become less anomalous as time progresses by including a second fractional time derivative acting on the term of diffusion. This paper introduces a radial basis function-generated finite difference (RBF-FD) method for solving the governing problem. The Grunwald–Letnikov formula with first-order accuracy is implemented to discretize the problem in the time direction, and the spatial variable is discretized using the local RBF-FD method. The convergence and stability of the time discretization scheme are deduced in an appropriate Sobolev space. The data distribution pattern within the support domain is considered to have a constant number of points. The numerical results on regular and irregular domains show the efficiency and high accuracy of the method and confirm the theoretical prediction.

A semidefinite programming approach for solving fractional optimal control problems

This paper presents a numerical scheme for solving fractional optimal control. The fractional derivative in this problem is in the Riemann–Liouville sense. The proposed method, based upon the method of moments, converts the fractional optimal control problem to a semidefinite optimization problem; namely, the nonlinear optimal control problem is converted to a convex optimization problem. The Grunwald–Letnikov formula is also used as an approximation for fractional derivative. The solution of fractional optimal control problem is found by solving the semidefinite optimization problem. Finally, numerical examples are presented to show the performance of the method.

Ameliorating the FOPID (PIλDμ) Controller Parameters for Heating Furnace using Optimization Techniques

Since the heating furnace system has emanated it has faced the problem of high power consumption, colossal amount of time to heat the substances and the vulnerability of getting exploded thus the objective of the paper is to achieve a system for same with less power consumption, whit time to heat the substances and making it safe from explosion. Using the mathematical way of modeling the dynamic critical systems the heating furnace is being modeled by using the damping, spring and mass elements. The integer order model of the system is being achieved by the Laplace transform and fractional order model for the same is obtained using the Grunwald-Letnikov formula. The Cohen-Coon tuning technique is being amalgamated with the Nelder-Mead, Interior-Point, Active-Set and Sequential Quadratic Programming optimization techniques respectively so as to design the FOPID controller for heating furnace. When the feedback systems were being formed then the outputs demonstrated that the system now consists the properties of less power consumption, less time to heat the substances along with less overshoot. Earlier the integer order model had the settling time (time taken to heat the substance), steady state error (power consumption) and overshoot (explosion) of 1500 seconds, 50% and 0% respectively. When the PID controller was designed for the same using Cohen-Coon tuning technique and forming a feedback system it had setting time of around 800 sec. and also the steady state error was brought to 0% but the overshoot went up to 35%. Therefore FOPID controller is being designed using the concocted technique that is the amalgamation of tuning technique and optimization techniques and forming and feedback system with FOM of heating furnace, the system yielded steady state error as 0%, where the settling time have been reduced to 300 seconds and overshoot between 7%-12%. Using the concocted technique that is the amalgamation of Cohen-Coon tuning technique with the optimization tuning techniques the FOPID controller was being formed for the FOM of the heating furnace which is being kept in feedback so as to form a system. Thus systems formed ameliorated the settling time i.e. time taken to heat the substance, the overshoot i.e. the vulnerability of getting exploded also remains low and the steady state error i.e. power consumption is also reduced drastically.

A GENERAL FRACTIONAL FINITE ELEMENT FORMULATION

Differential equations governing the behaviors of systems can be obtained through minimizations of energy functionals. In many applications, finite element formulations for a system can be developed directly using energy or some other types of functionals. One of the advantages of these formulations is that they do not require the natural boundary conditions to be imposed explicitly. In recent years, there has been a growing interest in the area of fractional variational calculus. Recently developed fractional variational formulations allow one to obtain the necessary Euler-Lagrange equations and the natural boundary conditions. However, closed form solutions for these problems may not be available, and a numerical technique may be necessary.In this paper, we present a general fractional finite element formulation for a class of fractional variational problems defined in terms of the Riemann-Liouville fractional derivatives. Specifically, we consider a functional quadratic in function y and its fractional derivative, where α is the order of the derivative. In this formulation, the domain of the integral is divided into several elements, and y and over the elements are written in terms of nodal variables and some shape functions. Various choices of shape functions are being considered. Here we approximate y and over each element using a linear function and a constant function, respectively. The constant function considered here represents at the center of the element. These approximations are used to reduce the original functional into a discretized quadratic form. Two approaches are presented to approximate at the center of an element. In the first scheme, 1) the number of node points is doubled; 2) the function values at the new node points are computed using the linear function; and 3) is expressed using Grunwald-Letnikov formula. In the second scheme, the Grunwald-Letnikov formula is modified to write at the center of an element in terms of nodal functional values.This modified Grunwald-Letnikov formula, when written for α = 1, gives the central difference formula for the derivative. These approximations are substituted into the above quadratic equation, the resulting equation is minimized, and the terminal kinematic conditions are imposed to obtain a set of algebraic equations. These algebraic equations can be solved using a direct or an iterative scheme. Here we use a direct scheme. Solution of these equations provides the response of the system.To demonstrate the applications of the formulations developed here, two fractional variational problems are solved using the scheme developed here, and the results are compared with analytical solutions and the solutions obtained using other schemes. Results show that 1) solutions converge as the number of node points is increased. However, in some cases, this convergence may be slow. 2) As α approaches 1, the classical solutions are recovered as expected.For integer order systems, finite element is a well established technique, and it is widely being used in academia and industries. It is hoped that this research will initiate a similar course for fractional order system.