Damped Oscillatory Solutions Research Articles

Euler’s computational approach for damped oscillatory solutions of RLC circuits

In this paper novel techniques for nonlinear frequency domain region problems with respect to linearity are examined. A notable methodology called as Modified Euler’s method and improved Euler’s method to perform nonlinear recurrence domain investigations of RLC circuit problems. Here a combination of waves approximates the driving forces to attain its damped oscillatory solutions. The two different cases were analyzed as, stability of the RLC circuit when the applied voltage is equal to zero and the forced oscillations of the RLC circuit when the applied voltage is equal to ?0?????. To keen the strength of investigation, stability analysis and non-oscillatory behavior of the RLC circuit are explained along with periodical time.

Qualitative analysis and approximate damped oscillatory solutions for a kind of nonlinear dispersive-dissipative equation

This paper makes qualitative analysis to the bounded traveling wave solutions for a kind of nonlinear dispersive-dissipative equation, and considers its solving problem. The relation is investigated between behavior of its solution and the dissipation coefficient. Further, all approximate damped oscillatory solutions when dissipation coefficient is small are presented by utilizing the method of undetermined coefficients according to the theory of rotated vector field in planar dynamical systems. Finally, error estimate is given by establishing the integral equation which reflects the relation between approximate and exact damped oscillatory solutions applying the idea of homogenization principle.

Bounded traveling wave solutions for MKdV-Burgers equation with the negative dispersive coefficient

This paper studies the bounded traveling wave solutions of MKdV-Burgers equation with the negative dispersive coefficient by the theory of planar dynamical systems, undetermined coefficients method. The global phase portraits under the different parameter conditions, as well as the existent number and conditions of the bounded traveling wave solutions are obtained for the dynamical system corresponding to the traveling wave solutions of MKdV-Burgers equation. The relation is investigated between the profiles of the bounded traveling wave solutions and dissipative coefficient. And a critical value characterizing the scale of dissipative effect, is given, which is different from the one proposed by R.F. Bikbaev in his article. Focusing on the open issue proposed by R.F. Bikbaev, based on the bell and kink profile solitary wave solutions of MKdV-Burgers equation we presented, approximate damped oscillatory solutions of MKdV-Burgers equation are obtained according to the evolution relation of orbits corresponding to the approximate damped oscillatory solutions in the global phase portraits.

Approximate damped oscillatory solutions and error estimates for the perturbed Klein–Gordon equation

The influence of perturbation on traveling wave solutions of the perturbed Klein–Gordon equation is studied by applying the bifurcation method and qualitative theory of dynamical systems. All possible approximate damped oscillatory solutions for this equation are obtained by using undetermined coefficient method. Error estimates indicate that the approximate solutions are meaningful. The results of numerical simulations also establish our analysis.

Bounded traveling wave solutions of variant boussinesq equation with a dissipation term and dissipation effect

This article studies bounded traveling wave solutions of variant Boussinesq equation with a dissipation term and dissipation effect on them. Firstly, we make qualitative analysis to the bounded traveling wave solutions for the above equation by the theory and method of planar dynamical systems, and obtain their existent conditions, number, and general shape. Secondly, we investigate the dissipation effect on the shape evolution of bounded traveling wave solutions. We find out a critical value r* which can characterize the scale of dissipation effect, and prove that the bounded traveling wave solutions appear as kink profile waves if |r|≥r*; while they appear as damped oscillatory waves if |r|≥r*. We also obtain kink profile solitary wave solutions with and without dissipation effect. On the basis of the above discussion, we sensibly design the structure of the approximate damped oscillatory solutions according to the orbits evolution relation corresponding to the component u(ξ) in the global phase portraits, and then obtain the approximate solutions (u˜(ξ),H˜(ξ)). Furthermore, by using homogenization principle, we give their error estimates by establishing the integral equation which reflects the relation between exact and approximate solutions. Finally, we discuss the dissipation effect on the amplitude, frequency, and energy decay of the bounded traveling wave solutions.

Qualitative analysis to traveling wave solutions of Zakharov–Kuznetsov–Burgers equation and its damped oscillatory solutions

The theory of planar dynamical systems is applied in this paper to carry out a qualitative analysis to the planar dynamical system corresponding to the bounded traveling wave solution of the Zakharov–Kuznetsov–Burgers equation, and obtain the existence and uniqueness of the bounded traveling wave solutions. According to the discussions on the relationships between the shapes of bounded traveling wave solutions and the dissipation coefficient d, a critical value d0 is found for arbitrary traveling wave speed v and integral constant g. This equation has a unique monotone kink profile solitary wave solution as the dissipation coefficient d satisfies |d¯|>d0; while it has a unique damped oscillatory solution as |d¯|<d0. This paper also presents the exact bell profile solitary wave solution as d=0. Furthermore, we appropriately design the structure of the damped oscillatory solution in light of the evolution relationships of the solution orbit in the global phase portraits to which the damped oscillatory solution corresponds and obtain its approximate solution by means of the undetermined coefficients method. Finally, based on the integral equation that reflects the relationships between the approximate damped oscillatory solution and its exact solution, the error estimate is given for the approximate damped oscillatory solution. The error is infinitesimal decreasing in exponential form.

A Class of Approximate Damped Oscillatory Solutions to Compound KdV-Burgers-Type Equation with Nonlinear Terms of Any Order: Preliminary Results

This paper is focused on studying approximate damped oscillatory solutions of the compound KdV-Burgers-type equation with nonlinear terms of any order. By the theory and method of planar dynamical systems, existence conditions and number of bounded traveling wave solutions including damped oscillatory solutions are obtained. Utilizing the undetermined coefficients method, the approximate solutions of damped oscillatory solutions traveling to the left are presented. Error estimates of these approximate solutions are given by the thought of homogeneous principle. The results indicate that errors between implicit exact damped oscillatory solutions and approximate damped oscillatory solutions are infinitesimal decreasing in the exponential form.

Shape analysis and damped oscillatory solutions for a class of nonlinear wave equation with quintic term

This paper aims at analyzing the shapes of the bounded traveling wave solutions for a class of nonlinear wave equation with a quintic term and obtaining its damped oscillatory solutions. The theory and method of planar dynamical systems are used to make a qualitative analysis to the planar dynamical system which the bounded traveling wave solutions of this equation correspond to. The shapes, existent number, and conditions are presented for all bounded traveling wave solutions. The bounded traveling wave solutions are obtained by the undetermined coefficients method according to their shapes, including exact expressions of bell and kink profile solitary wave solutions and approximate expressions of damped oscillatory solutions. For the approximate damped oscillatory solution, using the homogenization principle, its error estimate is given by establishing the integral equation, which reflects the relation between the exact and approximate solutions. It can be seen that the error is infinitesimal decreasing in the exponential form.

Qualitative Analysis and Travelling Wave Solutions for the Chaffee–Infante Equation

This paper employs the theory of planar dynamical systems and undetermined coefficient method to study travelling wave solutions to the Chaffee–Infante equation. By qualitative analysis, global phase portraits of the dynamic system corresponding to the equation are obtained for different parameter conditions. Furthermore, the relations between the properties of travelling wave solutions and the diffusion coefficient λ of the equation are investigated. In addition, all possible kink profile solitary wave solutions and approximate damped oscillatory solutions to the equation are obtained by using undetermined coefficient method. Error estimates indicate that the approximate solutions are meaningful. Based on these studies, the main contribution in this paper is to reveal the diffusion effect on travelling wave solutions to the Chaffee–Infante equation.

Shape Analysis and Solution to a Class of Nonlinear Wave Equation with Cubic Term

AbstractIn this paper, we analyze the relation between the shape of the bounded traveling wave solutions and dissipation coefficient of nonlinear wave equation with cubic term by the theory and method of planar dynamical systems. Two critical values which can characterize the scale of dissipation effect are obtained. If dissipation effect is not less than a certain critical value, the traveling wave solutions appear as kink profile; while if it is less than this critical value, they appear as damped oscillatory. All expressions of bounded traveling wave solutions are presented, including exact expressions of bell and kink profile solitary wave solutions, as well as approximate expressions of damped oscillatory solutions. For approximate damped oscillatory solution, using homogenization principle, we give its error estimate by establishing the integral equation which reflects the relations between the exact and approximate solutions. It can be seen that the error is an infinitesimal decreasing in the exponential form.

Qualitative analysis to the traveling wave solutions of Kakutani-Kawahara equation and its approximate damped oscillatory solution

In this paper, we apply the theory of planar dynamical systems tomake a qualitative analysis to the traveling wave solutions ofnonlinear Kakutani-Kawahara equation$u_t+uu_x+bu_{x x x}-a(u_t+uu_x)_x=0$ ($b>0, a\ge0$) and obtain theexistent conditions of the bounded traveling wave solutions. Indispersion-dominant case, we find that the unique bounded travelingwave solution of this equation has not only oscillatory property butalso damped property. Furthermore, according to the evolution oforbits in the global phase portraits, we present an approximatedamped oscillatory solution for this equation by the undeterminedcoefficients method. Finally, by the idea of homogenizationprinciples, we obtain an integral equation which reflects therelation between this approximate damped oscillatory solution andits exact solution, thereby having the error estimate. The error isan infinitesimal decreasing in exponential form. From the results inthis paper, it can be seen that the damped oscillatory solution ofKakutani-Kawahara equation in dispersion-dominant case still remainssome properties of solitary wave solution for KdV equation.

Approximate damped oscillatory solutions for compound KdV-Burgers equation and their error estimates

In this paper, we focus on studying approximate solutions of damped oscillatory solutions of the compound KdV-Burgers equation and their error estimates. We employ the theory of planar dynamical systems to study traveling wave solutions of the compound KdV-Burgers equation. We obtain some global phase portraits under different parameter conditions as well as the existence of bounded traveling wave solutions. Furthermore, we investigate the relations between the behavior of bounded traveling wave solutions and the dissipation coefficient r of the equation. We obtain two critical values of r, and find that a bounded traveling wave appears as a kink profile solitary wave if |r| is greater than or equal to some critical value, while it appears as a damped oscillatory wave if |r| is less than some critical value. By means of analysis and the undetermined coefficients method, we find that the compound KdV-Burgers equation only has three kinds of bell profile solitary wave solutions without dissipation. Based on the above discussions and according to the evolution relations of orbits in the global phase portraits, we obtain all approximate damped oscillatory solutions by using the undetermined coefficients method. Finally, using the homogenization principle, we establish the integral equations reflecting the relations between exact solutions and approximate solutions of damped oscillatory solutions. Moreover, we also give the error estimates for these approximate solutions.

Shape analysis of bounded traveling wave solutions and solution to the generalized Whitham-Broer-Kaup equation with dissipation terms

This paper deals with the problem of the bounded traveling wave solutions’ shape and the solution to the generalized Whitham-Broer-Kaup equation with the dissipation terms which can be called WBK equation for short. The authors employ the theory and method of planar dynamical systems to make comprehensive qualitative analyses to the above equation satisfied by the horizontal velocity component u(ξ) in the traveling wave solution (u(ξ), H(ξ)), and then give its global phase portraits. The authors obtain the existent conditions and the number of the solutions by using the relations between the components u(ξ) and H(ξ) in the solutions. The authors study the dissipation effect on the solutions, find out a critical value r*, and prove that the traveling wave solution (u(ξ),H(ξ)) appears as a kink profile solitary wave if the dissipation effect is greater, i.e., |r| ≥ r*, while it appears as a damped oscillatory wave if the dissipation effect is smaller, i.e., |r| < r*. Two solitary wave solutions to the WBK equation without dissipation effect is also obtained. Based on the above discussion and according to the evolution relations of orbits corresponding to the component u(ξ) in the global phase portraits, the authors obtain all approximate damped oscillatory solutions (ũ(ξ), \(\tilde H\)(ξ)) under various conditions by using the undetermined coefficients method. Finally, the error between the approximate damped oscillatory solution and the exact solution is an infinitesimal decreasing exponentially.

Bounded damped oscillatory solutions of Fisher equation

To research the bounded damped oscillatory solutions of nonlinear evolution equation, we choose the Fisher equation as an example. The solutions with negative amplitudes of Fisher equation may become meaningful in the context of nonscalar models describing excitable media (e.g. the Belousov-Zhabotinsky (BZ) reaction). The theory of planar dynamical systems is used to study the existence conditions of bounded traveling wave solutions of Fisher equation. The bounded approximate damped oscillatory analytic solution is given by using LS method and linearization method. And its error is also estimated.

Approximate Damped Oscillatory Solutions for Generalized KdV‐Burgers Equation and Their Error Estimates

We focus on studying approximate solutions of damped oscillatory solutions of generalized KdV‐Burgers equation and their error estimates. The theory of planar dynamical systems is employed to make qualitative analysis to the dynamical systems which traveling wave solutions of this equation correspond to. We investigate the relations between the behaviors of bounded traveling wave solutions and dissipation coefficient, and give two critical values λ1 and λ2 which can characterize the scale of dissipation effect, for right and left‐traveling wave solution, respectively. We obtain that for the right‐traveling wave solution if dissipation coefficient α ≥ λ1, it appears as a monotone kink profile solitary wave solution; that if 0 &lt; α &lt; λ1, it appears as a damped oscillatory solution. This is similar for the left‐traveling wave solution. According to the evolution relations of orbits in the global phase portraits which the damped oscillatory solutions correspond to, we obtain their approximate damped oscillatory solutions by undetermined coefficients method. By the idea of homogenization principle, we give the error estimates for these approximate solutions by establishing the integral equations reflecting the relations between exact and approximate solutions. The errors are infinitesimal decreasing in the exponential form.

Qualitative analysis and solutions of bounded traveling wave for B-BBM equation

In this paper, we apply the theory of planar dynamical systems to carry out qualitative analysis for the dynamical system corresponding to B-BBM equation, and obtain global phase portraits under various parameter conditions. Then, the relations between the behaviors of bounded traveling wave solutions and the dissipation coefficient µ are investigated. We find that a bounded traveling wave solution appears as a kink profile solitary wave solution when µ is more than the critical value obtained in this paper, while a bounded traveling wave solution appears as a damped oscillatory solution when µ is less than it. Furthermore, we explain the solitary wave solutions obtained in previous literature, and point out their positions in global phase portraits. In the meantime, approximate damped oscillatory solutions are given by means of undetermined coefficients method. Finally, based on integral equations that reflect the relations between the approximate damped oscillatory solutions and the implicit exact damped oscillatory solutions, error estimates for the approximate solutions are presented.

Exact travelling wave solutions for the dissipative (2 + 1)-dimensional AKNS equation

This paper employs the theory of planar dynamical systems and undetermined coefficient method to study travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation. By qualitative analysis, global phase portraits of the dynamic system corresponding to the equation are obtained under different parameter conditions. Furthermore, the relations between the properties of travelling wave solutions and the dissipation coefficient r of the equation are investigated. In addition, the possible bell profile solitary wave solution, kink profile solitary wave solutions and approximate damped oscillatory solutions of the equation are obtained by using undetermined coefficient method. Error estimates indicate that the approximate solutions are meaningful. Based on above studies, a main contribution in this paper is to reveal the dissipation effect on travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation.

Qualitative analysis and solutions of bounded travelling waves for the fluidized-bed modelling equation

We apply the theory of planar dynamical systems to carry out a qualitative analysis for the planar dynamical system corresponding to the fluidized-bed modelling equation. We obtain the global phase portraits of this system under various parameter conditions and the existence conditions of bounded travelling-wave solutions of this equation. According to the discussion on relationships between the behaviours of bounded travelling-wave solutions and the dissipation coefficients ε and δ, we find a critical value λ0 for arbitrary travelling-wave velocity υ. This equation has a unique damped oscillatory solution as ∥ε + δυ∥ &lt; λ0 and ∥ε + δυ∥ ≠ 0, while it has a unique monotone kink profile solitary-wave solution as ∥ε + δυ∥ &gt; λ0. By means of the undetermined coefficients method, we obtain the exact bell profile solitary-wave solution and monotone kink profile solitary-wave solution. Meanwhile, we obtain the approximate damped oscillatory solution. We point out the positions of these solutions in the global phase portraits. Finally, based on integral equations that reflect the relationships between the approximate damped oscillatory solutions and the implicit exact damped oscillatory solutions, error estimates for the approximate damped oscillatory solutions are presented.

Exact travelling wave solutions of the dissipative coupled Korteweg-de Vries equation

This paper employs the theory of planar dynamical systems and undetermined coefficient method to study travelling wave solutions of the dissipative coupled Korteweg-de Vries equation. The possible kink profile solitary wave solutions and approximate damped oscillatory solutions of the equation are obtained by using undetermined coefficient method. Error estimates indicate that the approximate solutions are meaningful.

Qualitative Analysis and Travelling Wave Solutions for the Generalized Pochhammer-Chree Equation with a Dissipation Term

The purpose of this paper is to reveal the influence of dissipation on travelling wave solutions of the generalized Pochhammer–Chree equation with a dissipation term, and provides travelling wave solutions for this equation. Applying the theory of planar dynamical systems, we obtain ten global phase portraits of the dynamic system corresponding to this equation under various parameter conditions. Moreover, we present the relations between the properties of travelling wave solutions and the dissipation coefficient r of this equation. We find that a bounded travelling wave solution appears as a bell profile solitary wave solution or a periodic travelling wave solution when r= 0; a bounded travelling wave solution appears as a kink profile solitary wave solution when |r| > 0 is large; a bounded travelling wave solution appears as a damped oscillatory solution when |r| > 0 is small. Further, by using undetermined coefficient method, we get all possible bell profile solitary wave solutions and approximate damped oscillatory solutions for this equation. Error estimates indicate that the approximate solutions are meaningful.