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In the past ** Abdul-Majid Wazwaz** has collaborated on articles with

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AbstractA singularly perturbed model problem with multiple distinct regular singular points is studied. The uniform approximation of Wazwaz and Hanson (1986) is effectively extended to many singular points, thus establishing a generalized version of that theorem where the classic inner and outer expansions are not employed. A leading order general asymptotic solution is correctly represented by a set of matched exponential asymptotic expansions, where each approximation contains dominant and recessive terms. The resonance criteria due to the influence of multiple singular points are discussed. For an even number of singular points, the eigenvalues were found to be positive with a minimal eigenvalue. However, negative eigenvalues with a maximal eigenvalue arise for an odd number of singular points. The maximal and the minimal eigenvalue may not be unique and each is 0(1).

AbstractA modified third order Runge-Kutta method for solving initial value problems of the form y1 = f(x,y) is presented. A formula is constructed by using an averaging of the functional values of the form (GM)2AM, where GM is the geometric mean averaging and AM is the arithmetic mean averaging. The technique was numerically tested, and the result shows a smaller error than the new method of [1] and [2], where they employed the geometric mean (GM) averaging only.

AbstractA modified fourth order Runge–Kutta method based on the combined effects of the arithmetic mean averaging (AM) and the geometric mean averaging (GM) of the functional values is developed. The approach is also applied to establish alternative numerical integration algorithms for the trapezoidal formula, Simpson's rule and Romberg integration. Numerical computations are carried out to compare the results of the modified methods with the results of the standard formulae.

AbstractThe purpose of this work is to introduce new types of sequences, whose terms are infinite series instead of real numbers, and to examine the convergence/divergence of these sequences and of the sum of its terms.

AbstractThis paper presents a physical model described by the (2+1)-dimensional nonlinear Schrödinger equation with variable coefficients (2D-VcNLSE). The 2D-VcNLSE is related to many physical phenomena in nonlinear optical fibers, Bose-Einstein condensates, and water waves. We study new types of nonautonomous complex wave solutions in the presence of inhomogeneous media. Various structures of these solutions such as bright and dark soliton and similarity solutions are investigated. Furthermore, different exact solutions for the 2D-VcNLSE are obtained via the G′/G-expansion method. Through 3D- and contour plots, we show that the dynamical behaviors of the obtained solutions can be effectively controlled by modulating the values of the arbitrary functions in these solutions.

AbstractWe operate variational iteration method for obtaining exact analytical solutions for nonlinear Schrödinger (NLS) equation with normal dispersive regimes. A set of diverse types of solitons that contains optical dark solitons are furnished with significant physical perspective. VIM gives the solution in a rapidly convergent series structure. The current analysis under study depicts pertinent attributes of considered method. Additionally, improved G′G expansion method is employed to NLS equation, generating soliton solutions having rich spatial-temporal configuration along with abundant free parameters.

AbstractA construction of compact and noncompact solutions for nonlinear dispersive partial differential equations of even order is considered. Two model equations of even orders k⩾4 are adopted to carry out this study. The analysis reveals the presence of distinct roots of different indices that play a significant role in the solutions.

AbstractThe effect of the order of nonlinear dispersive partial differential equation on the compact and noncompact solutions is studied in this paper. Two variants of the KdV equation of odd orders are examined. Our analysis reveals that the physical nature of these nonlinear KdV-type of equations are different. Two distinct sets of general formulas for each type of variants are performed to present a fairly complete understanding of the compact and noncompact structures.

AbstractIn this paper we study compacton structures for nonlinear dispersive equations. We show that the purely nonlinear dispersive equations, where the nonlinear dispersion interact with nonlinear convection, generate compacton solutions: solitons free of exponential wings. We also show that the defocusing branches generate solitary patterns solutions.

AbstractIn this paper, we propose a new approach to develop several new exact solutions to a fast nonlinear diffusion equation. This approach rests on the differential constraints obtained by the method of linear determining equations. The method is capable of reducing the nonlinear diffusion equation to systems of nonlinear ordinary equations. The proposed scheme is presented in a general way so that it can be used in other diffusion processes.

AbstractIn this paper, two mathematical variants of Kuramoto–Sivashinsky equation are examined. A framework is developed for studying these variants in one and higher dimensions. Two distinct sets of general formulae for each variant are developed. Compactons: solitons with compact support, and noncompact dispersive structures are formally established.

AbstractIn this paper, we develop a framework to obtain exact solutions to Fisher's equation and to a nonlinear diffusion equation of the Fisher type by employing Adomian decomposition method. The proposed scheme is supported by examining nonlinear diffusion equations of the Fisher type.

AbstractIn this work, the nonlinear equation K(m, n) is studied for all possible values of m and n. We show that this equation may exhibit compactons, solitons or periodic solutions. The analysis reveals the change of the physical structure of the solutions as a result of the change of m and n.

AbstractIn this paper, Adomian decomposition method is used to obtain analytical solution for the time-dependent Emden–Fowler type of equations and wave-type equation with singular behavior. The advantage of this single global method is employed to present a reliable framework is used to overcome the singularity behavior at the point x = 0 for both models. The proposed scheme reveals quite a number of remarkable features that will be helpful for identical problems. The work is supported by analyzing few examples where the convergence of the results is observed.

AbstractGeneralized Boussinesq type of equations with positive and negative exponents are examined. The analysis depends mainly on the sine–cosine ansatz. It is formally shown that these nonlinear models give rise to compactons, solitary patterns, solitons, and periodic solutions depending on the exponents and the coefficients of the derivatives of u(x, t). The presented scheme reveals quite a number of remarkable features that will be helpful for identical problems.

AbstractIn this work we use the tanh method for traveling wave solutions of the sine-Gordon and the sinh-Gordon equations. Several exact solutions of distinct physical structures are obtained. The method is powerful with minimal algebra work and is demonstrated for four models.

AbstractGeneralized forms of the KdV, the nonlinear heat conduction, and Burgers–Fisher equations are investigated. The analysis rests mainly on the standard tanh method. The focus is on the case where the parameter M is noninteger. A variety of exact travelling wave solutions of distinct physical structures are formally derived.

AbstractA variety of exact solutions for the (2 + 1) dimensional ZK–BBM equation are developed by means of the tanh method and the sine–cosine methods. Generalized forms of the ZK–BBM equation are studied. The tanh and the sine–cosine methods are reliable to derive solutions of distinct physical structures: compactons, solitons, solitary patterns and periodic solutions.

AbstractIn this paper, we present a reliable combination of Adomian decomposition algorithm and Padé approximants to investigate the Flierl–Petviashivili (FP) equation and its variants. The approach introduces an alternative framework designed to overcome the difficulty of the singular point at x = 0. We also investigate two generalized variants of the FP equation. The proposed framework reveals quite a number of remarkable features of the combination of the two algorithms.

AbstractIn this work, the generalized KdV equation with two power nonlinearities is studied. The tanh method and two sets of ansatze involving hyperbolic functions are introduced for analytic study of this equation. New kinds of solitary wave solutions are formally derived.

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