A numerical technique for solving fractional variational problems

This paper presents an accurate numerical method for solving a class of fractional variational problems (FVPs). The fractional derivative in these problems is in the Caputo sense. The proposed method is called fractional Chebyshev finite difference method. In this technique, we approximate FVPs and end up with a finite‐dimensional problem. The method is based on the combination of the useful properties of Chebyshev polynomials approximation and finite difference method. The Caputo fractional derivative is replaced by a difference quotient and the integral by a finite sum. The fractional derivative approximation using Clenshaw and Curtis formula introduced here, along with Clenshaw and Curtis procedure for the numerical integration of a non‐singular functions and the Rayleigh–Ritz method for the constrained extremum, is considered. By this method, the given problem is reduced to the problem for solving a system of algebraic equations, and by solving this system, we obtain the solution of FVPs. Special attention is given to study the convergence analysis and evaluate an error upper bound of the obtained approximate formula. Illustrative examples are included to demonstrate the validity and applicability of the proposed technique. A comparison with another method is given. Copyright © 2012 John Wiley & Sons, Ltd.     

A numerical technique for solving a class of fractional variational problems

This paper presents a numerical method for solving a class of fractional variational problems (FVPs) with multiple dependent variables, multi order fractional derivatives and a group of boundary conditions. The fractional derivative in the problem is in the Caputo sense. In the presented method, the given optimization problem reduces to a system of algebraic equations using polynomial basis functions. An approximate solution for the FVP is achieved by solving the system. The choice of polynomial basis functions provides the method with such a flexibility that initial and boundary conditions can be easily imposed. We extensively discuss the convergence of the method and finally present illustrative examples to demonstrate validity and applicability of the new technique.     

Variational iteration method for solving the space‐ and time‐fractional KdV equation

This paper presents numerical solutions for the space‐ and time‐fractional Korteweg–de Vries equation (KdV for short) using the variational iteration method. The space‐ and time‐fractional derivatives are described in the Caputo sense. In this method, general Lagrange multipliers are introduced to construct correction functionals for the problems. The multipliers in the functionals can be identified optimally via variational theory. The iteration method, which produces the solutions in terms of convergent series with easily computable components, requiring no linearization or small perturbation. The numerical results show that the approach is easy to implement and accurate when applied to space‐ and time‐fractional KdV equations. The method introduces a promising tool for solving many space–time fractional partial differential equations. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

A Combination of Variational and Penalty Methods for Solving a Class of Fractional Optimal Control Problems

This paper develops an approximate method, based on the combination of epsilon penalty and variational methods, for solving a class of multidimensional fractional optimal control problems. The fractional derivative is in the Caputo sense. In the presented method, utilizing the epsilon method, the given optimal control problem transforms into an unconstrained optimization problem; then, the equivalent variational equality is derived for the given unconstrained problem. The variational equality is approximately solved by applying a spectral method.     

The fractional variational iteration method improved with the Adomian series

In this work, an improved version of the fractional variational iteration method is presented, for solving fractional initial value problems. The nonlinear terms of fractional differential equations are linearized via the famous Adomian series. The fractional differential functions are employed in the numerical simulation. Two examples are given as illustrations.     

General fractional variational problem depending on indefinite integrals

In this report, we consider two kind of general fractional variational problem depending on indefinite integrals include unconstrained problem and isoperimetric problem. These problems can have multiple dependent variables, multiorder fractional derivatives, multiorder integral derivatives and boundary conditions. For both problems, we obtain the Euler-Lagrange type necessary conditions which must be satisfied for the given functional to be extremum. Also, we apply the Rayleigh-Ritz method for solving the unconstrained general fractional variational problem depending on indefinite integrals. By this method, the given problem is reduced to the problem for solving a system of algebraic equations using shifted Legendre polynomials basis functions. An approximate solution for this problem is obtained by solving the system. We discuss the analytic convergence of this method and finally by some examples will be showing the accurately and applicability for this technique.     

A Generalization of Ritz-Variational Method for Solving a Class of Fractional Optimization Problems

This paper presents an approximate method for solving a class of fractional optimization problems with multiple dependent variables with multi-order fractional derivatives and a group of boundary conditions. The fractional derivatives are in the Caputo sense. In the presented method, first, the given optimization problem is transformed into an equivalent variational equality; then, by applying a special form of polynomial basis functions and approximations, the variational equality is reduced to a simple linear system of algebraic equations. It is demonstrated that the derived linear system has a unique solution. We get an approximate solution for the initial optimization problem by solving the final linear system of equations. The choice of polynomial basis functions provides a method with such flexibility that all initial and boundary conditions of the problem can be easily imposed. We extensively discuss the convergence of the method and, finally, present illustrative test examples to demonstrate the validity and applicability of the new technique.     

Applying a Power Penalty Method to Numerically Pricing American Bond Options

In this paper, we aim to develop a numerical scheme to price American options on a zero-coupon bond based on a power penalty approach. This pricing problem is formulated as a variational inequality problem (VI) or a complementarity problem (CP). We apply a fitted finite volume discretization in space along with an implicit scheme in time, to the variational inequality problem, and obtain a discretized linear complementarity problem (LCP). We then develop a power penalty approach to solve the LCP by solving a system of nonlinear equations. The unique solvability and convergence of the penalized problem are established. Finally, we carry out numerical experiments to examine the convergence of the power penalty method and to testify the efficiency and effectiveness of our numerical scheme.     

Numerical comparison of methods for solving linear differential equations of fractional order

In this article, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving linear differential equations of fractional order. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. This paper will present a numerical comparison between the two methods and a conventional method such as the fractional difference method for solving linear differential equations of fractional order. The numerical results demonstrates that the new methods are quite accurate and readily implemented.     

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.     

A variational method using fractional order Hilbert spaces for tomographic reconstruction of blurred and noised binary images

We provide in this article a refined functional analysis of the Radon operator restricted to axisymmetric functions, and show that it enjoys strong regularity properties in fractional order Hilbert spaces. This study is motivated by a problem of tomographic reconstruction of binary axially symmetric objects, for which we have available one single blurred and noised snapshot. We propose a variational approach to handle this problem, consisting in solving a minimization problem settled in adapted fractional order Hilbert spaces. We show the existence of solutions, and then derive first order necessary conditions for optimality in the form of optimality systems.     

Numerical solution of nonlinear partial quadratic integro‐differential equations of fractional order via hybrid of block‐pulse and parabolic functions

In this paper, an effective numerical approach based on a new two‐dimensional hybrid of parabolic and block‐pulse functions (2D‐PBPFs) is presented for solving nonlinear partial quadratic integro‐differential equations of fractional order. Our approach is based on 2D‐PBPFs operational matrix method together with the fractional integral operator, described in the Riemann–Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations, which greatly simplifies the problem. By using Newton's iterative method, this system is solved, and the solution of fractional nonlinear partial quadratic integro‐differential equations is achieved. Convergence analysis and an error estimate associated with the proposed method is obtained, and it is proved that the numerical convergence order of the suggested numerical method is O(h³) . The validity and applicability of the method are demonstrated by solving three numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the exact solutions much easier.     

Epsilon-Ritz Method for Solving a Class of Fractional Constrained Optimization Problems

In this paper, epsilon and Ritz methods are applied for solving a general class of fractional constrained optimization problems. The goal is to minimize a functional subject to a number of constraints. The functional and constraints can have multiple dependent variables, multiorder fractional derivatives, and a group of initial and boundary conditions. The fractional derivative in the problem is in the Caputo sense. The constrained optimization problems include isoperimetric fractional variational problems (IFVPs) and fractional optimal control problems (FOCPs). In the presented approach, first by implementing epsilon method, we transform the given constrained optimization problem into an unconstrained problem, then by applying Ritz method and polynomial basis functions, we reduce the optimization problem to the problem of optimizing a real value function. The choice of polynomial basis functions provides the method with such a flexibility that initial and boundary conditions can be easily imposed. The convergence of the method is analytically studied and some illustrative examples including IFVPs and FOCPs are presented to demonstrate validity and applicability of the new technique.     

Variational formulation and efficient implementation for solving the tempered fractional problems

Because of the finiteness of the life span and boundedness of the physical space, the more reasonable or physical choice is the tempered power‐law instead of pure power‐law for the CTRW model in characterizing the waiting time and jump length of the motion of particles. This paper focuses on providing the variational formulation and efficient implementation for solving the corresponding deterministic/macroscopic models, including the space tempered fractional equation and time tempered fractional equation. The convergence, numerical stability, and a series of variational equalities are theoretically proved. And the theoretical results are confirmed by numerical experiments.     

Identification of constitutive parameters for fractional viscoelasticity

This paper develops a numerical model to identify constitutive parameters in the fractional viscoelastic field. An explicit semi-analytical numerical model and a finite difference (FD) method based numerical model are derived for solving the direct homogenous and regionally inhomogeneous fractional viscoelastic problems, respectively. A continuous ant colony optimization (ACO) algorithm is employed to solve the inverse problem of identification. The feasibility of the proposed approach is illustrated via the numerical verification of a two-dimensional identification problem formulated by the fractional Kelvin–Voigt model, and the noisy data and regional inhomogeneity etc. are taken into account.     

A new Jacobi operational matrix: An application for solving fractional differential equations

In this paper, we derived the shifted Jacobi operational matrix (JOM) of fractional derivatives which is applied together with spectral tau method for numerical solution of general linear multi-term fractional differential equations (FDEs). A new approach implementing shifted Jacobi operational matrix in combination with the shifted Jacobi collocation technique is introduced for the numerical solution of nonlinear multi-term FDEs. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. The proposed methods are applied for solving linear and nonlinear multi-term FDEs subject to initial or boundary conditions, and the exact solutions are obtained for some tested problems. Special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods.     

Fractional Variational Approach for Dissipative Mechanical Systems

More recently, a variational approach has been proposed by Lin and Wang for damping motion with a Lagrangian holding the energy term dissipated by a friction force. However, the modified Euler-Lagrange equation obtained within their for- malism leads to an incorrect Newtonian equation of motion due to the nonlocality of the Lagrangian. In this communication, we generalize this approach based on the fractional actionlike variational approach and we show that under some simple restric- tions connected to the fractional parameters introduced in the fractional formalism, this problem may be solved.     

Numerical solutions of time fractional degenerate parabolic equations by variational iteration method with Jumarie‐modified Riemann–Liouville derivative

In this article, the fractional variational iteration method is employed for computing the approximate analytical solutions of degenerate parabolic equations with fractional time derivative. The time‐fractional derivatives are described by the use of a new approach, the so‐called Jumarie modified Riemann–Liouville derivative, instead in the sense of Caputo. The approximate solutions of our model problem are calculated in the form of convergent series with easily computable components. Moreover, the numerical solution is compared with the exact solution and the quantitative estimate of accuracy is obtained. The results of the study reveal that the proposed method with modified fractional Riemann–Liouville derivatives is efficient, accurate, and convenient for solving the fractional partial differential equations in multi‐dimensional spaces without using any linearization, perturbation or restrictive assumptions. Copyright © 2011 John Wiley & Sons, Ltd.     

Power Penalty Method for a Linear Complementarity Problem Arising from American Option Valuation

In this paper, we present a power penalty function approach to the linear complementarity problem arising from pricing American options. The problem is first reformulated as a variational inequality problem; the resulting variational inequality problem is then transformed into a nonlinear parabolic partial differential equation (PDE) by adding a power penalty term. It is shown that the solution to the penalized equation converges to that of the variational inequality problem with an arbitrary order. This arbitrary-order convergence rate allows us to achieve the required accuracy of the solution with a small penalty parameter. A numerical scheme for solving the penalized nonlinear PDE is also proposed. Numerical results are given to illustrate the theoretical findings and to show the effectiveness and usefulness of the method. This work was partially supported by a research grant from the University of Western Australia and the Research Grant Council of Hong Kong, Grants PolyU BQ475 and PolyU BQ493.     

Variational Approach to Homoclinic Solutions for Fractional Hamiltonian Systems

In this paper, we discuss the existence and multiplicity of homoclinic solutions for fractional Hamiltonian systems with left and right Liouville–Weyl fractional derivatives. Sufficient conditions ensuring the existence of an unbounded sequence of homoclinic solutions for the given problem are obtained via variational approach.