In this paper, we present a general formulation for a fractional optimal control problem (FOCP), in which the state and co-state equations are given in terms of the left fractional derivatives. We develop the forward–backward sweep method (FBSM) using the Adams-type predictor–corrector method to solve the FOCP. We present a fractional model for transmission dynamics of human immunodeficiency virus/acquired immunodeficiency syndrome (HIV/AIDS) with treatment and incorporate three control efforts (effective use of condoms, ART treatment and behavioral change control) into the model aimed at controlling the spread of HIV/AIDS epidemic. The necessary conditions for fractional optimal control of the disease are derived and analyzed. The numerical results show that implementing all the control efforts increases the life time and the quality of life those living with HIV and decreases significantly the number of HIV-infected and AIDS people. Also, the maximum levels of the controls and the value of objective functional decrease when the derivative order [Formula: see text] limits to 1 ([Formula: see text]). In addition, the effect of the fractional derivative order [Formula: see text] ([Formula: see text]) on the spread of HIV/AIDS epidemic and the treatment of HIV-infected population is investigated. The results show that the derivative order [Formula: see text] can play the role of using ART treatment in the model.

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In this work, we present travelling wave solutions for the Burgers, Burgers-Huxley and modified Burgers-KdV equations. The (G /G)-expansion method is used to determine travelling wave solutions of these sets of equations. The travelling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. It is shown that the proposed method is direct, effective and can be used for many other nonlinear evolution equations in mathematical physics.

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In this paper, analytic solutions of the variant Boussinesq equations are obtained by the homotopy analysis and the homotopy Padé methods. The obtained approximation using homotopy method contains an auxiliary parameter, which is a simple way to control and adjust the convergence region and rate of solution series. The approximation solutions by [m,m] homotopy Padé technique is often independent of auxiliary parameter ℏ, and this technique accelerates the convergence of the related series. Copyright © 2013 John Wiley & Sons, Ltd.