Time-dependent Variable Coefficients Research Articles

Exact Solutions for the Modified Burgers Equation with Additional Time-dependent Variable Coefficient

In this article, we investigated new travelling wave solutions for the modified Burgers equation with additional time-dependent variable coefficient via the functional variable method. The performance of this method is reliable and effective and gives the exact solitary wave solutions. All solutions of this equation have been examined and three dimensional graphics of the obtained solutions have been drawn by using the Matlab program. The exact solutions have its great importance to reveal the internal mechanism of the physical phenomena. This method presents a wider applicability for handling nonlinear wave equations.

New Soliton Solutions of the Burgers Equation with Additional Time-dependent Variable Coefficient

In this article, we use the functional variational method to solve the Burgers equation with an additional time-dependent variable coefficient. The main advantage of the proposed method over other methods is that it allows to obtain more new solutions of the equation. Among the solutions obtained, new soliton solutions should be noted, which are of great importance for revealing the internal mechanism of physical phenomena. Three-dimensional graphs of solutions are constructed using the mathematical program Matlab. For a better understanding of the physical properties of some of the resulting solutions, their graphical representations are shown. This method is effective for finding exact solutions to many other similar wave equations.

Classical symmetries of the Klein–Gordon–Zakharov equations with time-dependent variable coefficients

In this article, we employ the group-theoretic methods to explore the Lie symmetries of the Klein–Gordon–Zakharov equations, which include time-dependent coefficients. We obtain the Lie point symmetries admitted by the Klein–Gordon–Zakharov equations along with the forms of variable coefficients. From the resulting symmetries, we construct similarity reductions.The similarity reductions are further analyzed using the power series method/approach and furnished the series solutions. Additionally, the convergence of the series solutions has been reported.

Analytical solutions for a class of variable-order fractional Liu system under time-dependent variable coefficients

In this article, we present a nonlinear model of the Liu system that includes fractional derivatives of variable-order. Due to the nonlocality of the dynamical system, we introduce the fractional derivative with power laws, exponential decay laws, and generalized Mittag-Leffler functions as kernels. We provide a detailed analysis of the existence and uniqueness of the proposed model, as well as the stability of these equations. Due to the existence of time-varying fractional derivatives, the proposed variable-order fractional system exhibits more complex characteristics and more degrees of freedom than an integer or conventional constant fractional-order chaotic Liu oscillator. Different chaotic behaviors can be obtained by using different smooth functions defined within the interval (0,1] as variable orders for fractional derivatives in the simulations. Furthermore, simulations demonstrate that fractional chaotic systems with variable orders can be synchronized.

Multiple soliton solutions of the Korteweg-de Vries in the frame of the time-dependent variable coefficient

Multiple soliton solutions of the Korteweg-de Vries in the frame of the time-dependent variable coefficient

Multiple Soliton Solutions for Coupled Modified Korteweg–de Vries (mkdV) with a Time-Dependent Variable Coefficient

In this manuscript, we implement analytical multiple soliton wave and singular soliton wave solutions for coupled mKdV with a time-dependent variable coefficient. Based on the similarity transformation and Hirota bilinear technique, we construct both multiple wave kink and wave singular kink solutions for coupled mKdV with a time-dependent variable coefficient. We implement the Hirota bilinear technique to compute analytical solutions for the coupled mKdV system. Such calculations are made by using a software with symbolic computation software, for instance, Maple. Recently some researchers used Maple in order to show that the bilinear method of Hirota is a straightforward technique which can be used in the approach of differential, nonlinear models. We analyzed whether the experiments proved that the procedure is effective and can be successfully used for many other mathematical models used in physics and engineering. The results of this study display that the profiles of multiple-kink and singular-kink soliton types can be efficiently controlled by selecting the particular form of a similar time variable. The changes in the solitons based on the changes in the arbitrary function of time allows for more applications of them in applied sciences.

A Lie group treatment on a generalized evolution Fisher type equation with variable coefficients

In this study, a general second-order evolution equation of the Fisher-type, with time-dependent variable coefficients, is considered. This equation contains many well-known equations, and obtained results may be applicable in investigating other evolution equations. Lie symmetries and corresponding invariant solutions of the considered problem are studied by a non-traditional Lie symmetry method (LSM). Prolongation of the model is an essential part of the Lie symmetry method, which in the current work, we analyze by the heir-equations. Finally, different types of solutions depending on the variable coefficients, such as the exponential solutions, trigonometric solutions, Bessel solutions, and Mathieu solutions, are extracted.

Finite Time Blow Up of Solutions for the m-Laplacian Equation With Variable Coefficients

In this work, we deal with the m- Laplacian equation with time dependent variable coefficients. Under suitable conditions on variable coefficients, we prove the blow up of solutions for finite time with negative initial energy. These results partially generalize and extend some recent ones in previous literature.

Stabilization of semilinear wave equations with time-dependent variable coefficients and memory

Stabilization of semilinear wave equations with time-dependent variable coefficients and memory

Painlevé integrability and new soliton solutions for (2 + 1)-dimensional Bogoyavlensky–Konopelchenko equation and generalized Bogoyavlensky–Konopelchenko equation with variable coefficients in fluid mechanics

The time-dependent variable coefficients of Bogoyavlensky–Konopelchenko (BK) equation and generalized Bogoyavlensky–Konopelchenko (gBK) equation are considered in this paper. The integrability test by Painlevé analysis is being implemented on both the considered equations. An auto-Bäcklund transformation has been generated with the help of Painlevé analysis for both equations. Auto-Bäcklund transformation method has been used for obtaining the analytic solutions. By using auto-Bäcklund transformation method, three different analytic solution families have been derived for each of the considered equations. Multi-soliton solutions are also calculated for both the considered equations by using Hereman and Nuseir algorithm. All the results are expressed graphically in 3D by varying different functions and parametric values. These graphs reveal the physical significance of equations under consideration.

A Unified Lattice Boltzmann Model for Fourth Order Partial Differential Equations with Variable Coefficients.

In this work, a unified lattice Boltzmann model is proposed for the fourth order partial differential equation with time-dependent variable coefficients, which has the form . A compensation function is added to the evolution equation to recover the macroscopic equation. Applying Chapman-Enskog expansion and the Taylor expansion method, we recover the macroscopic equation correctly. Through analyzing the error, our model reaches second-order accuracy in time. A series of constant-coefficient and variable-coefficient partial differential equations are successfully simulated, which tests the effectiveness and stability of the present model.

New abundant analytic solutions for generalized KdV6 equation with time-dependent variable coefficients using Painlevé analysis and auto-Bäcklund transformation

This paper considers the generalized KdV6 equation with time-dependent variable coefficients. The integrability of the considered equation is being examined by the Painlevé analysis method. Further, an auto-Bäcklund transformation method has been adopted to obtain the analytic solutions. Using this technique, five novel families of analytic solutions in the form of rational, exponential, hyperbolic and trigonometric functions have been successfully found for the considered equation. New kink-antikink and periodic soliton solutions have been discovered using this method. The solutions are graphically depicted to show the physical significance of the problem under consideration.

Lie Symmetries, Closed-Form Solutions, and Various Dynamical Profiles of Solitons for the Variable Coefficient (2+1)-Dimensional KP Equations

This investigation focuses on two novel Kadomtsev–Petviashvili (KP) equations with time-dependent variable coefficients that describe the nonlinear wave propagation of small-amplitude surface waves in narrow channels or large straits with slowly varying width and depth and non-vanishing vorticity. These two variable coefficients, Kadomtsev–Petviashvili (VCKP) equations in (2+1)-dimensions, are the main extensions of the KP equation. Applying the Lie symmetry technique, we carry out infinitesimal generators, potential vector fields, and various similarity reductions of the considered VCKP equations. These VCKP equations are converted into nonlinear ODEs via two similarity reductions. The closed-form analytic solutions are achieved, including in the shape of distinct complex wave structures of solitons, dark and bright soliton shapes, double W-shaped soliton shapes, multi-peakon shapes, curved-shaped multi-wave solitons, and novel solitary wave solitons. All the obtained solutions are verified and validated by using back substitution to the original equation through Wolfram Mathematica. We analyze the dynamical behaviors of these obtained solutions with some three-dimensional graphics via numerical simulation. The obtained variable coefficient solutions are more relevant and useful for understanding the dynamical structures of nonlinear KP equations and shallow water wave models.

Space-Time Fractional KdV–Burger–Kuramato Equation with Time Dependent Variable Coefficients: Lie Symmetry, Explicit Power Series Solution, Convergence Analysis and Conservation Laws

Space-Time Fractional KdV–Burger–Kuramato Equation with Time Dependent Variable Coefficients: Lie Symmetry, Explicit Power Series Solution, Convergence Analysis and Conservation Laws

Painlevé analysis, auto-Bäcklund transformation and new exact solutions of [formula omitted] and [formula omitted]-dimensional extended Sakovich equation with time dependent variable coefficients in ocean physics

This article considers time-dependent variable coefficients (2+1) and (3+1)-dimensional extended Sakovich equation. Painlevé analysis and auto-Bäcklund transformation methods are used to examine both the considered equations. Painlevé analysis is appeared to test the integrability while an auto-Bäcklund transformation method is being presented to derive new analytic soliton solution families for both the considered equations. Two new family of exact analytical solutions are being obtained successfully for each of the considered equations. The soliton solutions in the form of rational and exponential functions are being depicted. The results are also expressed graphically to illustrate the potential and physical behaviour of both equations. Both the considered equations have applications in ocean wave theory as they depict new solitary wave soliton solutions by 3D and 2D graphs.

Global existence of solutions for the semilinear Klein-Gordon equation with a time-dependent variable coefficient

The semilinear Klein-Gordon equation with initial conditions is studied in de Sitter spacetime. The L? decay estimates are derived for the solutions to the linear Klein- Gordon equations with and without source term in de Sitter spacetime. It is also showed the global existence of solutions to the initial value problem with power type non-linear terms for small initial data by using these estimates

A study of multi-soliton solutions, breather, lumps, and their interactions for kadomtsev-petviashvili equation with variable time coeffcient using hirota method

This paper investigates the new KP equation with variable coefficients of time ‘t’, broadly used to elucidate shallow water waves that arise in plasma physics, marine engineering, ocean physics, nonlinear sciences, and fluid dynamics. In 2020, Wazwaz [1] proposed two extensive KP equations with time-variable coefficients to obtain several soliton solutions and used Painlevé test to verify their integrability. In light of the research described above, we chose one of the integrated KP equations with time-variable coefficients to obtain multiple solitons, rogue waves, breather waves, lumps, and their interaction solutions relating to the suitable choice of time-dependent coefficients. For this KP equation, the multiple solitons and rogue waves up to fourth-order solutions, breather waves, and lump waves along with their interactions are achieved by employing Hirota's method. By taking advantage of Wolfram Mathematica, the time-dependent variable coefficient's effect on the newly established solutions can be observed through the three-dimensional wave profiles, corresponding contour plots. Some newly formed mathematical results and evolutionary dynamical behaviors of wave-wave interactions are shown in this work. The obtained results are often more advantageous for the analysis of shallow water waves in marine engineering, fluid dynamics, and dusty plasma, nonlinear sciences, and this approach has opened up a new way to explain the dynamical structures and properties of complex physical models. This study examines to be applicable in its influence on a wide-ranging class of nonlinear KP equations.

On the existence and uniqueness of weak solutions to time-fractional elliptic equations with time-dependent variable coefficients

This paper is devoted to discussing the existence and uniqueness of weak solutions to time-fractional elliptic equations having time-dependent variable coefficients. To obtain the main result, our strategy is to combine the Galerkin method, a basic inequality for the fractional derivative of convex Lyapunov candidate functions, the Yoshida approximation sequence and the weak compactness argument.

Lie Symmetry Analysis of the Nonlinear Schrödinger Equation with Time Dependent Variable Coefficients

In this work, we study the generalized nonlinear Schrodinger equation having time dependent variable coefficients by using Lie group method. We find the Lie point symmetries admitted by the Schrodinger equation for arbitrary variable coefficients and from the obtained symmetries, we constructed the similarity reductions. As the further symmetry analysis of the reduced ordinary differential equations was not possible due to lack of symmetries, we, therefore, solve one of the reduced nonlinear ordinary differential equation for completeness by using power series method and presented the exact series solution. The convergence of the series solutions is also established.

Space time fractional Drinfel'd-Sokolov-Wilson system with time-dependent variable coefficients: Symmetry analysis, power series solutions and conservation laws

In this work we aim to apply the Lie-symmetry method via the Riemann-Liouville fractional derivative based on continuous group of transformations to examine the symmetry reduction of space time fractional Drinfel'd-Sokolov-Wilson (DSW) system with time-dependent variable coefficients. The reduced Nonlinear fractional ordinary differential equations (NLFODEs) with variable coefficients are further studied for the exact solution by using the power series method. The exact solutions obtained in form of power series and their convergence show the accuracy and efficiency of the proposed method, which is simple and accurate in comparison to other methods to find the exact solution of nonlinear fractional partial differential equations (NLFPDEs). Also, the new conservation theorem and Noether's operators are used to construct conservation laws of the governing system.