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This paper presents efficient numerical techniques for solving fractional optimal control problems (FOCP) based on orthonormal wavelets. These wavelets are like Legendre wavelets, Chebyshev wavelets, Laguerre wavelets and Cosine And Sine (CAS) wavelets. The formulation of FOCP and properties of these wavelets are presented. The fractional derivative considered in this problem is in the Caputo sense. The performance index of FOCP has been considered as function of both state and control variables and the dynamic constraints are expressed by fractional differential equation. These wavelet methods are applied to reduce the FOCP as system of algebraic equations by applying the method of constrained extremum which consists of adjoining the constraint equations to the performance index by a set of undetermined Lagrange multipliers. These algebraic systems are solved numerically by Newton's method. Illustrative examples are discussed to demonstrate the applicability and validity of the wavelet methods. [ABSTRACT FROM AUTHOR]

In this paper, new exact solutions of fractional nonlinear acoustic wave equations have been devised. The travelling periodic wave solutions of fractional Burgers–Hopf equation and Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation have obtained by first integral method. Nonlinear ultrasound modelling is found to have an increasing number of applications in both medical and industrial areas where due to high pressure amplitudes the effects of nonlinear propagation are no longer negligible. Taking nonlinear effects into account, the ultrasound beam analysis makes more accurate in these applications. The Burgers–Hopf equation is one of the extensively studied models in mathematical physics. In addition, the KZK parabolic nonlinear wave equation is one of the most widely employed nonlinear models for propagation of 3D diffraction sound beams in dissipative media. In the present analysis, these nonlinear equations have solved by first integral method. As a result, new exact analytical solutions have been obtained first time ever for these fractional order acoustic wave equations. The obtained results are presented graphically to demonstrate the efficiency of this proposed method. [ABSTRACT FROM AUTHOR]

In this paper, efficient numerical techniques have been proposed to solve nonlinear Hammerstein integral equation arising from chemical phenomenon. The following integral equation forms the basis for the conductor like screening model for real solvent which appeared in chemical phenomenon. Our aim is to solve the Hammerstein nonlinear integral equation μ S ( σ ) = − R T ln [ ∫ P S ( σ ′ ) exp ( − E i n t ( σ , σ ′ ) − μ S ( σ ′ ) R T ) d σ ′ ] where R is the gas constant, T is the temperature and the term E i n t ( σ , σ ′ ) denotes the interaction energy expression for the segments with screening charge density σ and σ ′ respectively, the molecular interaction in solvent is P S ( σ ) and the chemical potential of the surface segments is described by μ S ( σ ) which is to be determined. This COSMO-RS integral equation has been solved by Bernstein collocation method, Haar wavelet method, and Sinc collocation method. These methods reduce the integral equation to nonlinear system of algebraic equations and then this algebraic system has been solved numerically by Newton’s method. Comparison has been done for these methods. Illustrative examples have been discussed to demonstrate the validity and applicability of Bernstein collocation method, Haar wavelet method and Sinc collocation method. [ABSTRACT FROM AUTHOR]

In this paper, a novel approach comprising the modified decomposition method with Fourier transform has been implemented for the approximate solution of fractional sine-Gordon equation u tt - R D x α u + sin u = 0 where R D x α is the Riesz space fractional derivative, 1 ≤ α ≤ 2 . For α = 2, it becomes classical sine-Gordon equation u tt − u xx + sin u = 0 and corresponding to α = 1, it becomes nonlocal sine-Gordon equation u tt − Hu + sin u = 0 which arises in Josephson junction theory, where H is the Hilbert transform. The fractional sine-Gordon equation is considered as an interpolation between the classical sine-Gordon equation (corresponding to α = 2) and nonlocal sine-Gordon equation (corresponding to α = 1). Here the analytic solution of fractional sine-Gordon equation is derived by using the modified decomposition method with Fourier transform. Then, we analyze the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method. [ABSTRACT FROM AUTHOR]

In this article, the Legendre spectral collocation method has been applied to solve Fredholm integro-differential-difference equations with variable coefficients. The proposed method is based on the Gauss–Legendre points with the basis functions of Lagrange polynomials. Usually, this type of integral equations are very difficult to solve analytically as well as numerically. The presented method applied to the integral equation reduces to solve the system of algebraic equations. Also the numerical results obtained by Legendre spectral collocation method have been compared with the results obtained by existing methods. Illustrative examples have been discussed to demonstrate the validity and applicability of the presented method. [ABSTRACT FROM AUTHOR]

In the present paper, we construct the analytical exact solutions of a nonlinear evolution equation in mathematical physics, viz. time-fractional Kuramoto–Sivashinsky equation by a new proposed method via fractional complex transform. As a result, new types of exact analytical solutions are obtained. Here the fractional derivative is described in Jumarie’s modified Riemann–Liouville sense. [ABSTRACT FROM AUTHOR]

In this paper, the analytical approximate traveling wave solutions of Whitham-Broer-Kaup (WBK) equations, which contain blow-up solutions and periodic solutions, have been obtained by using the coupled fractional reduced differential transform method. By using this method, the solutions were calculated in the form of a generalized Taylor series with easily computable components. The convergence of the method as applied to the WBK equations is illustrated numerically as well as analytically. By using the present method, we can solve many linear and nonlinear coupled fractional differential equations. The results justify that the proposed method is also very efficient, effective and simple for obtaining approximate solutions of fractional coupled modified Boussinesq and fractional approximate long wave equations. Numerical solutions are presented graphically to show the reliability and efficiency of the method. Moreover, the results are compared with those obtained by the Adomian decomposition method (ADM) and variational iteration method (VIM), revealing that the present method is superior to others. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper presents an efficient numerical method for finding solutions of the nonlinear Fredholm integral equations system of second kind based on Bernstein polynomials basis. The numerical results obtained by the present method have been compared with those obtained by B-spline wavelet method. This proposed method reduces the system of integral equations to a system of algebraic equations that can be solved easily any of the usual numerical methods. Numerical examples are presented to illustrate the accuracy of the method. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

• A novel numerical approach for solving SIVIE has been proposed first time ever. • Operational matrices are used to transform the SIVIE into a system of algebraic equations. • Resultant system of algebraic equations has been solved by the collocation method. • This technique has the advantage of being simple to implement. • Error estimate, convergence, and stability analysis are also established. In this paper, a numerical method is implemented to solve the stochastic Itô-Volterra integral equations. In this approach, operational matrices have been applied to reduce the stochastic Itô-Volterra integral equations to linear algebraic equations. Then collocation method is applied to solve the algebraic equations. The error, convergence, and stability analysis of the proposed method are discussed. Also, the steps of the proposed method have been presented in the form of an algorithm. Numerical examples are introduced to confirm the efficiency and reliability of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this article dispersive optical soliton solutions that are governed by the time-fractional Schrödinger–Hirota equation have been presented. Some new soliton solutions of the time-fractional SH equations are obtained in this work. Using extended auxiliary equation method, optical soliton solutions are sought for the fractional SH equations with power law nonlinearity as well as Kerr law nonlinearity. Using fractional complex transform the time-fractional SH equation is converted into the nonlinear ordinary differential equation, and then, the resulting equation is solved using a novel analytical method viz. extended auxiliary equation method. As a result of this, some new optical soliton solutions have been successfully obtained. The obtained results show that the proposed method is a straightforward technique to find out new optical solutions of time-fractional SH equation. [ABSTRACT FROM AUTHOR]