Implementation of variational iteration method for various types of linear and nonlinear partial differential equations

There are various linear and nonlinear one-dimensional partial differential equations that are the focus of this research. There are a large number of these equations that cannot be solved analytically or precisely. The evaluation of nonlinear partial differential equations, even if analytical solutions exist, may be problematic. Therefore, it may be necessary to use approximate analytical methodologies to solve these issues. As a result, a more effective and accurate approach must be investigated and analyzed. It is shown in this study that the Lagrange multiplier may be used to get an ideal value for parameters in a functional form and then used to construct an iterative series solution. Linear and nonlinear partial differential equations may both be solved using the variational iteration method (VIM) method, thanks to its high computing power and high efficiency. Decoding and analyzing possible Korteweg-De-Vries, Benjamin, and Airy equations demonstrates the method’s ability. With just a few iterations, the produced findings are very effective, precise, and convergent to the exact answer. As a result, solving nonlinear equations using VIM is regarded as a viable option

The time-fractional mZK equation for gravity solitary waves and solutions using sech-tanh and radial basic function method

In recent years, we know that gravity solitary waves have gradually become the research spots and aroused extensive attention; on the other hand, the fractional calculus have been applied to the biology, optics and other fields, and it also has attracted more and more attention. In the paper, by employing multi-scale analysis and perturbation methods, we derive a new modified Zakharov–Kuznetsov (mZK) equation to describe the propagation features of gravity solitary waves. Furthermore, based on semi-inverse and Agrawal methods, the integer-order mZK equation is converted into the time-fractional mZK equation. In the past, fractional calculus was rarely used in ocean and atmosphere studies. Now, the study on nonlinear fluctuations of the gravity solitary waves is a hot area of research by using fractional calculus. It has potential value for deep understanding of the real ocean–atmosphere. Furthermore, by virtue of the sech-tanh method, the analytical solution of the time-fractional mZK equation is obtained. Next, using the above analytical solution, a numerical solution of the time-fractional mZK equation is given by using radial basis function method. Finally, the effect of time-fractional order on the wave propagation is explained. &nbsp

Traveling Wave Solutions of Space-Time Fractional Generalized Fifth-Order KdV Equation

The Korteweg-de Vries (KdV) equation, especially the fractional higher order one, provides a relatively accurate description of motions of long waves in shallow water under gravity and wave propagation in one-dimensional nonlinear lattice. In this article, the generalized exp⁡(-Φ(ξ))-expansion method is proposed to construct exact solutions of space-time fractional generalized fifth-order KdV equation with Jumarie’s modified Riemann-Liouville derivatives. At the end, three types of exact traveling wave solutions are obtained which indicate that the method is very practical and suitable for solving nonlinear fractional partial differential equations

A Computational Scheme for the Numerical Results of Time-Fractional Degasperis–Procesi and Camassa–Holm Models

This article presents an idea of a new approach for the solitary wave solution of the modified Degasperis–Procesi (mDP) and modified Camassa–Holm (mCH) models with a time-fractional derivative. We combine Laplace transform (LT) and homotopy perturbation method (HPM) to formulate the idea of the Laplace transform homotopy perturbation method (LHPTM). This study is considered under the Caputo sense. This proposed strategy does not depend on any assumption and restriction of variables, such as in the classical perturbation method. Some numerical examples are demonstrated and their results are compared graphically in 2D and 3D distribution. This approach presents the iterations in the form of a series solutions. We also compute the absolute error to show the effective performance of this proposed schemeThis research funded by Basque Government through Grant IT1155-22

Lie symmetry analysis, conservation laws and analytical solutions for chiral nonlinear Schrödinger equation in (2 + 1)-dimensions

In this work, we consider the chiral nonlinear Schrödinger equation in (2 + 1)-dimensions, which describes the envelope of amplitude in many physical media. We employ the Lie symmetry analysis method to study the vector field and the optimal system of the equation. The similarity reductions are analyzed by considering the optimal system. Furthermore, we find the power series solution of the equation with convergence analysis. Based on a new conservation law, we construct the conservation laws of the equation by using the resulting symmetries.&nbsp

Analytical and Numerical Methods for Differential Equations and Applications

The book is a printed version of the Special issue Analytical and Numerical Methods for Differential Equations and Applications, published in Frontiers in Applied Mathematics and Statistic

Structure and dynamics of solitary waves in fluid media

This research deals with the study of nonlinear solitary waves in fluid media. The equations which model surface and internal waves in fluids have been studied and used in this research. The approach to study the structure and dynamics of internal solitary waves in near-critical situations is the traditional theoretical and numerical study of nonlinear wave processes based on the methods of dynamical systems. The synergetic approach has been exploited, which presumes a combination of theoretical and numerical methods. All numerical calculations were performed with the desktop personal computer. Traditional and novel methods of mathematical physics were actively used, including Fourier analysis technique, inverse scattering method, Hirota method, phase-plane analysis, analysis of integral invariants, finite-difference method, Petviashvili and Yang–Lakoba numerical iterative techniques for the numerical solution of Partial Differential Equation. A new model equation, dubbed the Gardner–Kawahara equation, has been suggested to describe wave phenomena in the near-critical situations, when the nonlinear and dispersive coefficients become anomalously small. Such near-critical situations were not studied so far, therefore this study is very topical and innovative. Results obtained will shed a light on the structure of solitary waves in near-critical situation, which can occur in two-layer fluid with strong surface tension between the layers. A family of solitary waves was constructed numerically for the derived Gardner–Kawahara equation; their structure has been investigated analytically and numerically. The problem of modulation stability of quasi-monochromatic wave-trains propagating in a media has also being studied. The Nonlinear Schrödinger equation (NLSE) has been derived from the unidirectional Gardner–Ostrovsky equation and a general Shrira equation which describes both surface and internal long waves in a rotating fluid. It was demonstrated that earlier obtained results (Grimshaw & Helfrich, 2008; 2012; Whitfield & Johnson, 2015a; 2015b) on modulational stability/instability are correct within the limited range of wavenumbers where the Ostrovsky equation is applicable. In the meantime, results obtained in this Thesis and published in the paper (Nikitenkova et al., 2015) are applicable in the wider range of wavenumbers up to k = 0. It was shown that surface and internal oceanic waves are stable with respect to selfmodulation at small wavenumbers when k → 0 in contrast to what was mistakenly obtained in (Shrira, 1981). In Chapter 4 new exact solutions of the Kadomtsev-Petviashvili equation with a positive dispersion are obtained in the form of obliquely propagating skew lumps. Specific features of such lumps were studied in details. In particular, the integral characteristics of single lumps (mass, momentum components and energy) have been calculated and presented in terms of lump velocity. It was shown that exact stationary multi-lump solutions can be constructed for this equation. As the example, the exact bilump solution is presented in the explicit form and illustrated graphically. The relevance of skew lumps to the real physical systems is discussed

Mixed Chebyshev and Legendre polynomials differentiation matrices for solving initial-boundary value problems

A new form of basis functions structures has been constructed. These basis functions constitute a mix of Chebyshev polynomials and Legendre polynomials. The main purpose of these structures is to present several forms of differentiation matrices. These matrices were built from the perspective of pseudospectral approximation. Also, an investigation of the error analysis for the proposed expansion has been done. Then, we showed the presented matrices' efficiency and accuracy with several test functions. Consequently, the correctness of our matrices is demonstrated by solving ordinary differential equations and some initial boundary value problems. Finally, some comparisons between the presented approximations, exact solutions, and other methods ensured the efficiency and accuracy of the proposed matrices