K. Hosseini

New exact solutions of nonlinear conformable time-fractional Boussinesq equations using the modified Kudryashov method

Abstract In this paper, the nonlinear Boussinesq equations with the conformable time-fractional derivative are solved analytically using the well-established modified Kudryashov method. As a consequence, a number of new exact solutions for this type of equations are formally derived. It is believed that the method is one of the most effective techniques for extracting new exact solutions of nonlinear fractional differential equations.

Variational iteration method for solving Fokker-Planck equation

In this paper variational iteration method is implemented to solve linear and nonlinear Fokker–Planck equations, and some similar equations. To illustrate the simplicity and reliability of the method some examples are provided and comparisons are made between the variational iteration method and the Adomian decomposition method. Comparison shows that variational iteration method is more effective and convenient to use and overcomes the difficulty arising in calculating Adomian polynomials. Also, this method can successfully be applied to a large class of problems.

Bright and singular soliton solutions of the conformable time-fractional Klein–Gordon equations with different nonlinearities

Abstract Finding the exact solutions of nonlinear fractional differential equations has gained considerable attention, during the past two decades. In this paper, the conformable time-fractional Klein–Gordon equations with quadratic and cubic nonlinearities are studied. Several exact soliton solutions, including the bright (non-topological) and singular soliton solutions are formally extracted by making use of the ansatz method. Results demonstrate that the method can efficiently handle the time-fractional Klein–Gordon equations with different nonlinearities.

New exact traveling wave solutions of the Tzitzéica-type evolution equations arising in non-linear optics

Abstract The properties of Tzitzéica equations in non-linear optics have been the subject of many recent studies. In this article, a new and effective modification of Kudryashov method is adopted to study this class of non-linear evolution equations. As an achievement, new exact traveling wave solutions of Tzitzéica, Dodd–Bullough–Mikhailov (DBM) and Tzitzéica–Dodd–Bullough (TDB) equations are formally extracted. It is believed that the modified Kudryashov method along with the symbolic computation package suggests a promising technique to handle non-linear evolution equations in non-linear optics.

New exact solutions of the coupled sine-Gordon equations in nonlinear optics using the modified Kudryashov method

Abstract The main aim of this article is studying the coupled sine-Gordon equations in nonlinear optics, which describe the propagation of an optical pulse in fibre waveguide. New exact solutions of the coupled sine-Gordon equations have been derived through the use of the well-organized modified Kudryashov method.

New exact solutions of the Tzitzéica-type equations in non-linear optics using the expa function method

Abstract One specific class of non-linear evolution equations, known as the Tzitzéica-type equations, has received great attention from a group of researchers involved in non-linear science. In this article, new exact solutions of the Tzitzéica-type equations arising in non-linear optics, including the Tzitzéica, Dodd–Bullough–Mikhailov and Tzitzéica–Dodd–Bullough equations, are obtained using the expa function method. The integration technique actually suggests a useful and reliable method to extract new exact solutions of a wide range of non-linear evolution equations.

Optical soliton solutions of the cubic-quintic non-linear Schrödinger’s equation including an anti-cubic term

Abstract A wide range of problems in different fields of the applied sciences especially non-linear optics is described by non-linear Schrödinger’s equations (NLSEs). In the present paper, a specific type of NLSEs known as the cubic-quintic non-linear Schrödinger’s equation including an anti-cubic term has been studied. The generalized Kudryashov method along with symbolic computation package has been exerted to carry out this objective. As a consequence, a series of optical soliton solutions have formally been retrieved. It is corroborated that the generalized form of Kudryashov method is a direct, effectual, and reliable technique to deal with various types of non-linear Schrödinger’s equations.