Gardner Equation Research Articles

Solitary wave solutions to Gardner equation using improved tan$ \left(\frac{\Omega(\Upsilon)}{2}\right) $-expansion method

<abstract><p>In this study, the improved tan$ \left(\frac{\Omega(\Upsilon)}{2}\right) $-expansion method is used to construct a variety of precise soliton and other solitary wave solutions of the Gardner equation. Gardner equation is extensively utilized in plasma physics, quantum field theory, solid-state physics and fluid dynamics. It is the simplest model for the description of water waves with dual power law nonlinearity. Hyperbolic, exponential, rational and trigonometric traveling wave solutions are obtained. The retrieved solutions include kink solitons, bright solitons, dark-bright solitons and periodic wave solutions. The efficacy of this method is determined by the comparison of the newly obtained results with already reported results.</p></abstract>

ON THE GENERALIZED VARIATIONAL PRINCIPLE OF THE FRACTAL GARDNER EQUATION

The fractal calculus has gained more widespread attention in the last years. The fractal variational principle plays a major role in the fractal travelling wave theory of the fractal PDEs. This paper develops the fractal generalized variational principle (GVP) of the fractal Gardner equation by virtue of the Semi-inverse method (SIM) for the first time. On the other hand, we also discuss and verify the fractal GVP via the fractal two-scale transform (FTST) from another dimension field. The extracted fractal GVP shows the conservation laws through the energy form in the fractal space, and can be manipulated to explore the fractal solitary wave properties.

Head-on collision between two-counter-propagating electron acoustic soliton and double layer in an unmagnetized plasma

The head-on collision between two-counter-propagating electron acoustic solitons and double layers (DLs) in an unmagnetized collisionless multi-species plasma consisting of inertial cold electron fluid and (α, q)-distributed hot electrons and positrons has been analyzed with the stationary background of massive positive ions. For nonlinear analysis of colliding wave phenomena, the coupled Korteweg–de Vries equation (KdVE), modified KdVE (mKdVE), and standard Gardner equation have been derived by adopting the extended Poincaré–Lighthill–Kuo technique. The effect of non-dimensional parameters on the collisional KdV, mKdV, and Gardner solitons (GSs) and DLs has been examined in detail by considering the limiting cases of (α, q)-distributions. It is found that the plasma model supports (i) the compressive and rarefactive collisional KdV solitons and GSs, (ii) only compressive mKdV solitons, and (iii) only rarefactive collisional DLs. The rarefactive collisional solitons are more affected by nonextensivity and the increase of the temperature of electrons than their compressive counterpart, whereas the rarefactive collisional DLs only existed in the presence of nonthermality.

Simulation of generalized time fractional Gardner equation utilizing in plasma physics for non-linear propagation of ion-acoustic waves

In this work, radial basis function collocation method (RBFCM) is implemented for generalized time fractional Gardner equation (GTFGE). The RBFCM is meshless and easy-to-implement in complex geometries and higher dimensions, therefore, it is highly demanding. In this work, the Caputo derivative of fractional order ? ? (0, 1] is used to approximate the first order time derivative whereas, Crank-Nicolson scheme is hired to approximate space derivatives. The numerical solutions are presented and discussed, which demonstrate that the method is effective and accurate.

Application of Generalized Logistic Function to Travelling Wave Solutions for a Class of Nonlinear Evolution Equations

The generalized Logistic function that solves a first-order nonlinear ODE with an arbitrary positive power term of the dependent variable is introduced in this paper, by means of which the traveling wave solutions of a class of nonlinear evolution equations, including the generalized Fisher equation, the generalized Nagumo equation, the generalized Burgers-Fisher equation, the generalized Gardner equation, the generalized KdV-Burgers equation, and the generalized Benney equation, are obtained successfully. In these particular cases, traveling wave solutions of several important model PDEs in mathematical physics are also discovered.

Dynamical Behaviors of Lumpoff and Rogue Wave Solutions for Nonlocal Gardner Equation

In this paper, we got a novel kind of rogue waves with the predictability of (2 + 1)-dimensional nonlocal Gardner equation with the aid of Maple according to the Hirota bilinear model. We first construct a general quadratic function to derive the general lump solution for the mentioned equation. At the same time, the lumpoff solutions are demonstrated with more free autocephalous parameters, in which the lump solution is localized in all directions in space. Moreover, when the lump solution is cut by twin-solitons, special rogue waves are also introduced. Based on the data available in the literature, the resulting soliton solutions are innovative, developed, distinctive, and significant and can be applied to more complex phenomena, and they are immensely active for nonlinear models of classical and fractional-order type. To examine the dynamic behavior of the waves, contour 3D and 2D plots of several obtained findings are sketched by assigning specific values to the parameters. Furthermore, we obtain new sufficient solutions containing cross-kink, periodic-kink, multi-waves, and solitary wave solutions. It is worth noting that the emerging time and place of the rogue waves depend on the moving path of the lump solution.

Density Estimation from Compressional Velocity Using New Gardner Factors for Two Carbonate Formations, Amara Oil Field.

Bulk density (matrix density and pores fluid density) is a very important physical property that is used as main parameter to estimate physical characteristics (porosity, water and hydrocarbons saturation, shale volume and many others). The compensation from bulk density and sonic velocity is used to calculate acoustic impedance, reflection coefficient, and other elastic properties. Gardner equation have default values for (a) (factor) = 0.31 if Vp in (m/sec) and (b) (exponent) = 0.25 for Vp (m/sec) and these values change according to the lithology condition in the study area to give more accurate results, for this reason The object of the presented work is to calculate new (a) factor and (b) exponent to measure bulk density from compressional velocity using Gardner1974 equation for two formations from middle Cretaceous age, the data was taken from two open hole well logs Am-6 and Am-10 in Amara oil field. Techlog 2015 software was used to complete this work. The results showed that the new values for Mishrif Formation factor (a) were (1.21) in Am-6 and (1.32) in Am-10 and (b) exponent values were (0.46) in Am-6 and (0.40) in Am-10, while for Rumiala Formation factor (a) were (1.35) in Am-6 and (1.333) in Am-10 and (b) exponent value was (0.40) in both Am-6 and Am-10, these values give best fitting with minimum err with field measuring bulk density logs data and these values can used to measure bulk density log in the study area when field measuring bulk density log is not available.

Theoretical estimates of the parameters of longitudinal undular bores in polymethylmethacrylate bars based on their measured initial speeds

We study the evolution of the longitudinal release wave that is generated by induced tensile fracture as it propagates through solid rectangular polymethylmethacrylate (PMMA) bars of different constant cross-section. High-speed multi-point photoelasticity is used to register the strain wave at three distances from the fracture site in each experiment. In all cases, oscillations develop at the bottom of the release wave that exhibit the qualitative features of an undular bore. The pre-strain, post-strain, strain rate of the release wave and the cross-section dimensions determine the evolution of the oscillations. From the wave speed and strain rate close to the fracture site, we estimate the strain rate of the release wave as well as the growth of the amplitude and duration of the leading oscillation away from the fracture site by using formulae derived from the simple analytical solution of the linearized Gardner equation (linearized near the pre-strain level at fracture). Our estimates are then compared to experimental data, where good agreements of these three parameters are found between the predictions of the model and the experimental observations. Thus, we developed an approach to estimating the key characteristics of the developing unsteady undular bore based on the measured initial speeds of the longitudinal and shear waves. This does not require a prior knowledge of the elastic moduli for the conditions of the experiments, which in PMMA are known to be strain rate dependent.

Effective Analytical Computational Technique for Conformable Time-Fractional Nonlinear Gardner Equation and Cahn-Hilliard Equations of Fourth and Sixth Order Emerging in Dispersive Media

The aim of this paper is to investigate the approximate solutions of nonlinear temporal fractional models of Gardner and Cahn-Hilliard equations. The fractional models of the Gardner and Cahn-Hilliard equations play an important role in pulse propagation in dispersive media. The time-fractional derivative is observed in the conformable framework. In this orientation, a reliable computationally algorithm is designed and developed by following a residual error and multivariable power series expansion. Basically, the approximate solutions of pulse wave function of the fractional higher-order Gardner and Cahn-Hilliard equations are obtained in the form of a conformable convergent fractional series. Relevant consequences are theoretically and numerically investigated under the conformable sense. Besides, the analysis of the error and convergence of the developed technique are discussed. Some of the unidirectional homogeneous physical applications of the posed models in a finite compact regime are tested to confirm the theoretical aspects, demonstrate different evolutionary dynamics, and highlight the superiority of the novel developed algorithm compared to other existing analytical methods. For this purpose, associated graphs are displayed in two and three dimensions. Growing and decaying modes of the fractional parameters are analyzed for several α values. From a numerical viewpoint, the simulations and results declare that the proposed iterative algorithm is indeed straightforward and appropriate with efficiency for long-wavelength solutions of nonlinear partial differential equations.

Rational Solitons in the Gardner-Like Models

Rational solutions of nonlinear evolution equations are considered in the literature as a mathematical image of rogue waves, which are anomalously large waves that occur for a short time. In this work, bounded rational solutions of Gardner-type equations (the extended Korteweg-de Vries equation), when a nonlinear term can be represented as a sum of several terms with arbitrary powers (not necessarily integer ones), are found. It is shown that such solutions describe first-order algebraic solitons, kinks, and pyramidal and table-top solitons. Analytical solutions are obtained for the Gardner equation with two nonlinear terms, the powers of which differ by a factor of 2. In other cases, the solutions are obtained numerically. Gardner-type equations occur in the description of nonlinear wave dynamics in a fluid layer with continuous or multilayer stratification, as well as in multicomponent plasma, and their solutions are used for the interpretation of rogue waves.

Implications of machine learning on geomechanical characterization and sand management: a case study from Hilal field, Gulf of Suez, Egypt

Sand production is one of the major challenges in the oil and gas industry, so a comprehensive geomechanical analysis is necessary to mitigate sand production in mature fields. As the pore pressure drastically decline in depleted reservoirs, the sand production risk becomes more critical and needs to be studied. However, the absence of key logs in many wells is a big challenge in the petroleum industry, and most geologists and engineers use empirical equations to predict missed log intervals. We conducted a comprehensive geomechanical modeling study on a full set of logs from two wells from the Hilal field, Gulf of Suez, Egypt, to infer the geomechanical elements and predict sand production. We have used the multi-arm calipers to calculate the actual depth of damage ratio to validate the geomechanical parameters in the prognosis model and confirm the stress orientations. We used machine learning approach to infer key sonic log in X-10 well to replace the empirical equations. The multi-arm calipers analysis showed an observed anisotropy in the hole diameter size with more enlargement in the ENE direction and fits with the minimum horizontal stress direction in the direction of N 60oE. The later also deduced the maximum horizontal stress direction in N150 ° based on the induced fractures from borehole image data in a nearby field. We developed and compared two sand management models: one using empirical equation and the other using machine learning. The model driven by the Gardner equation suggests sand production from day one, which is not matched with the production data, while the model driven by machine learning suggests no sand production risk, which is matched with the actual production data. Our results demonstrate the advantage of using machine learning technique in geomechanical studies on the classical empirical equations in the area of study that can be applied in other basins. The findings of this study can help with a better understanding of the implications of machine learning on geomechanical characterization and sand management.

APPLICATION OF THE EXTENDED F-EXPANSION METHOD FOR SOLVING THE FRACTIONAL GARDNER EQUATION WITH CONFORMABLE FRACTIONAL DERIVATIVE

In this study, we derive a new fractional Gardner equation with the conformable fractional derivative for the first time. Using the extended F-expansion method, different traveling wave solutions expressed in the form of the hyperbolic solutions such as sech, tanh, csch, coth, sinh and their combination, and the triangular solutions like csc, sec, tan, cot and their combination are obtained. Finally, the solutions are illustrated through the 3D plots. The results strongly prove that the proposed approach is effective and can help us understand the nonlinear problems arising in physics better.

New analytical solutions by the application of the modified double sub-equation method to the (1 + 1)-Schamel-KdV equation, the Gardner equation and the Burgers equation

In this research we present the application of the modified double sub-equation guess solution together with the analytical solutions of the Riccati equation to obtain new analytical exact solutions to the (1 + 1)-Schamel-KdV equation, the (1 + 1)-dimensional Gardner equation (or combined KdV-mKdV) and the nonlinear evolution (1 + 1)-dimensional Burgers equation. Results show some conditions between the allowed values of the interaction coefficients and the parameters of the allowed analytical solutions of the double sub-equation guess solution. The resulting new soliton solutions have by first time the proper real phase value behavior, not seen in previous applications of the double sub-equation method. Additionally, it has been revealed that handle the complexiton process with two different traveling variables have as an important consequence that the mixing between trigonometric and hyperbolic solutions is only observed for the analytical solutions of the nonlinear evolution (1 + 1)-dimensional Burgers equation, while no mixing between trigonometric and hyperbolic solutions is observed for the solutions of the (1 + 1)-Schamel-KdV ((1 + 1)-S-KdV) equation or the (1 + 1)-dimensional Gardner equation. Numerical simulations are provided in 3-D graphs and 2-D plots representing the general characteristics of the resulting analytical solutions.

The Application of the Functional Variable Method for Solving the Loaded Non-linear Evaluation Equations

In this article, we construct exact traveling wave solutions of the loaded Korteweg-de Vries, the loaded modified Korteweg-de Vries, and the loaded Gardner equation by the functional variable method. The performance of this method is reliable and effective and gives the exact solitary and periodic wave solutions. All solutions to these equations have been examined and 3D graphics of the obtained solutions have been drawn by using the Matlab program. We get some traveling wave solutions, which are expressed by the hyperbolic functions and trigonometric functions. The graphical representations of some obtained solutions are demonstrated to better understand their physical features, including bell-shaped solitary wave solutions, singular soliton solutions, and solitary wave solutions of kink type. Our results reveal that the method is a very effective and straightforward way of formulating the exact traveling wave solutions of non-linear wave equations arising in mathematical physics and engineering.

Effects of rotation and topography on internal solitary waves governed by the rotating Gardner equation

Abstract. Nonlinear oceanic internal solitary waves are considered under the influence of the combined effects of saturating nonlinearity, Earth's rotation, and horizontal depth inhomogeneity. Here the basic model is the extended Korteweg–de Vries equation that includes both quadratic and cubic nonlinearity (the Gardner equation) with additional terms incorporating slowly varying depth and weak rotation. The complicated interplay between these different factors is explored using an approximate adiabatic approach and then through numerical solutions of the governing variable depth, i.e., the rotating Gardner model. These results are also compared to analysis in the Korteweg–de Vries limit to highlight the effect of the cubic nonlinearity. The study explores several particular cases considered in the literature that included some of these factors to illustrate limitations. Solutions are made to illustrate the relevance of this extended Gardner model for realistic oceanic conditions.

Approximate solutions to the Gardner equation by spectral modified Exponential Time Differencing method

This manuscript is concerned with the numerical study of spatio-temporal solution profiles of the Gardner equation by implementing a well-known spectral modified numerical scheme based on the Exponential Time Differencing method proposed by Kassam and Trefethan. The modified scheme takes full advantage of contour integral method for improving the numerical stability. But it requires judicious selection of contour path with suitable discretization to minimize the error estimates for solving a particular nonlinear partial differential equation. The efficiency of the scheme is demonstrated by testing it on some test examples – especially measuring discrete maximum norm errors and global relative errors. The associated conservative laws are also numerically verified via attaining very insignificant deviation in the conserved quantities from the initial values even after a large simulation time.

Formation of Nonlinear Stationary Structures in Ionospheric Plasma

Solar radiation, along with cosmic radiation, ionizes the Earth's atmosphere and creates a dense layer known as the ionosphere. By considering weakly relativistic degenerate plasma in the planetary ionosphere, we have studied the formation and nature of the solitary structure, electrostatic double layers (DLs), and so on. As we have considered weak relativistic degeneracy, only electrons get accelerated and are the key to stationary structures. We have implemented Sagdeev's pseudopotential method, standard Gardner equation, and accordingly identified regimes where the solitary formation and DLs may be observed. We have studied the parametric influences on solitons and DLs. Furthermore, we extended our investigation to the oscillatory Rossby solitons in the ionospheric plasma. The results may help interpret many high-energy atmospheric observations in the ionospheric plasma.

A Mesh-Free Collocation Method Based on RBFs for the Numerical Solution of Hunter–Saxton and Gardner Equations

In this paper, the radial basis function (RBF) collocation method is applied to solve nonlinear partial differential equations (PDEs). First, the given equation is reduced to time-discrete form using Ө-weighted scheme. Then, with the help of RBFs, the given PDEs are transformed into a system of algebraic equations that is easy to solve. The proposed technique is verified by solving Hunter–Saxton and Gardner equations. For the solution of these problems, three types of RBFs, including multiquadric (MQ), inverse multiquadric (IMQ), and Gaussian (GA) have been used. The obtained solution is validated with the help of absolute error, L2, and L∞ error norms. The involved shape parameter is selected by the automatic algorithm that lies in the stable region of the method. The stability of the scheme is discussed by the spectral matrix method and validated computationally.

Analytical Investigation of Fractional-Order Cahn–Hilliard and Gardner Equations Using Two Novel Techniques

In this paper, we used the natural decomposition approach with non-singular kernel derivatives to find the solution to nonlinear fractional Gardner and Cahn–Hilliard equations arising in fluid flow. The fractional derivative is considered an Atangana–Baleanu derivative in Caputo manner (ABC) and Caputo–Fabrizio (CF) throughout this paper. We implement natural transform with the aid of the suggested derivatives to obtain the solution of nonlinear fractional Gardner and Cahn–Hilliard equations followed by inverse natural transform. To show the accuracy and validity of the proposed methods, we focused on two nonlinear problems and compared it with the exact and other method results. Additionally, the behavior of the results is demonstrated through tables and figures that are in strong agreement with the exact solutions.

Approximate Analytical and Numeric Solutions to a Forced Damped Gardner Equation.

In this paper, some exact traveling wave solutions to the integrable Gardner equation are reported. The ansatz method is devoted for deriving some exact solutions in terms of Jacobi and Weierstrass elliptic functions. The obtained analytic solutions recover the solitary waves, shock waves, and cnoidal waves. Also, the relation between the Jacobi and Weierstrass elliptic functions is obtained. In the second part of this work, we derive some approximate analytic and numeric solutions to the nonintegrable forced damped Gardner equation. For the approximate analytic solutions, the ansatz method is considered. With respect to the numerical solutions, the evolution equation is solved using both the finite different method (FDM) and cubic B-splines method. A comparison between different approximations is reported.