Chaudry Masood Khalique

Application of Legendre wavelets for solving fractional differential equations

In this paper, we develop a framework to obtain approximate numerical solutions to ordinary differential equations (ODEs) involving fractional order derivatives using Legendre wavelet approximations. The properties of Legendre wavelets are first presented. These properties are then utilized to reduce the fractional ordinary differential equations (FODEs) to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. Results show that this technique can solve the linear and nonlinear fractional ordinary differential equations with negligible error compared to the exact solution.

Rational solutions to an extended Kadomtsev-Petviashvili-like equation with symbolic computation

Associated with the prime number p = 3 , the generalized bilinear operators are adopted to yield an extended Kadomtsev-Petviashvili-like (eKP-like) equation. With symbolic computation, eighteen classes of rational solutions to the resulting eKP-like equation are generated from a search for polynomial solutions to the corresponding generalized bilinear equation.

A note on rational solutions to a Hirota-Satsuma-like equation

Abstract With the generalized bilinear operators based on a prime number p = 3 , a Hirota-Satsuma-like equation is proposed. Rational solutions are generated and graphically described by using symbolic computation software Maple.

A direct bilinear Bäcklund transformation of a (2+1)-dimensional Korteweg–de Vries-like model

Abstract We directly construct a bilinear Backlund transformation (BT) of a (2+1)-dimensional Korteweg–de Vries-like model. The construction is based on a so-called quadrilinear representation. The resulting bilinear BT is in accordance with the auxiliary-independent-variable-involved one derived with the Bell-polynomial scheme. Moreover, by applying the gauge transformation and the Hirota perturbation technique, multisoliton solutions are iteratively computed.

A new approach for solving a system of fractional partial differential equations

In this paper we propose a new method for solving systems of linear and nonlinear fractional partial differential equations. This method is a combination of the Laplace transform method and the Iterative method and here after called the Iterative Laplace transform method. This method gives solutions without any discretization and restrictive assumptions. The method is free from round-off errors and as a result the numerical computations are reduced. The fractional derivative is described in the Caputo sense. Finally, numerical examples are presented to illustrate the preciseness and effectiveness of the new technique.

Exact solutions of the (2+1 )-dimensional Zakharov-Kuznetsov modified equal width equation using Lie group analysis

This paper obtains the solution of the (2+1)-dimensional Zakharov-Kuznetsov modified equal width equation. The Lie group analysis is used to carry out the integration of this equation. The solutions obtained include the topological, non-topological soliton solution, cnoidal waves and the traveling wave solutions.

Application of the Laplace decomposition method for solving linear and nonlinear fractional diffusion―wave equations

Abstract In this paper, the Laplace decomposition method is employed to obtain approximate analytical solutions of the linear and nonlinear fractional diffusion–wave equations. This method is a combined form of the Laplace transform method and the Adomian decomposition method. The proposed scheme finds the solutions without any discretization or restrictive assumptions and is free from round-off errors and therefore, reduces the numerical computations to a great extent. The fractional derivative described here is in the Caputo sense. Some illustrative examples are presented and the results show that the solutions obtained by using this technique have close agreement with series solutions obtained with the help of the Adomian decomposition method.

Group analysis of KdV equation with time dependent coefficients

We study the generalized KdV equation having time dependent variable coefficients of the damping and dispersion from the Lie group-theoretic point of view. Lie group classification with respect to the time dependent coefficients is performed. The optimal system of one-dimensional subalgebras of the Lie symmetry algebras are obtained. These subalgebras are then used to construct a number of similarity reductions and exact group-invariant solutions, including soliton solutions, for some special forms of the equations.

Travelling wave solutions of nonlinear evolution equations using the simplest equation method

In this paper, the simplest equation method has been used for finding the exact solutions of three nonlinear evolution equations, namely the Vakhnenko-Parkes equation, the generalized regularized long wave equation and the symmetric regularized long wave equation. All three of these equations arise in fluids science, so finding their exact solutions is of great importance.

On the Solutions and Conservation Laws of a Coupled Kadomtsev-Petviashvili Equation

A coupled Kadomtsev-Petviashvili equation, which arises in various problems in many scientific applications, is studied. Exact solutions are obtained using the simplest equation method. The solutions obtained are travelling wave solutions. In addition, we also derive the conservation laws for the coupled Kadomtsev-Petviashvili equation.