Tanh Method Research Articles

Analytical and numerical techniques for initial‐boundary value problems of Kolmogorov–Petrovsky–Piskunov equation

AbstractIn this work, we are interested in finding exact and numerical solutions of well‐known Kolmogorov–Petrovsky–Piskunov (KPP) equation. Here, we introduce modified extended tanh method for finding the exact solutions of KPP equation and an average linear finite difference scheme for numerical investigations. It has been observed that the numerical scheme so employed is stable and accurate enough to produce stable results.

Symmetry reduction and new interaction solutions for the negative-order potential KdV equation

New soliton–cnoidal interaction solutions for the negative-order potential KdV equation are studied with the help of the consistent tanh expansion method. The non-local symmetry for the negative-order potential KdV equation is derived from the truncated Painlevé expansion method. The non-local symmetry is transformed into the standard Lie point symmetry by introducing new dependent variables and the finite symmetry transformation is also presented to construct new exact solutions with non-zero seed solution. The similarity solutions of the enlarged negative-order potential KdV system are obtained through different constant selections.

Construction of invariant solutions and conservation laws to the (2+1)-dimensional integrable coupling of the KdV equation

Under investigation in this paper is the (2+1)-dimensional integrable coupling of the KdV equation which has applications in wave propagation on the surface of shallow water. Firstly, based on the Lie symmetry method, infinitesimal generators and an optimal system of the obtained symmetries are presented. At the same time, new analytical exact solutions are computed through the tanh method. In addition, based on Ibragimov’s approach, conservation laws are established. In the end, the objective figures of the solutions of the coupling of the KdV equation are performed.

Lie Symmetry Analysis, Traveling Wave Solutions, and Conservation Laws to the (3 + 1)-Dimensional Generalized B-Type Kadomtsev-Petviashvili Equation

In this paper, the (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili(BKP) equation is studied applying Lie symmetry analysis. We apply the Lie symmetry method to the (3 + 1)-dimensional generalized BKP equation and derive its symmetry reductions. Based on these symmetry reductions, some exact traveling wave solutions are obtained by using the tanh method and Kudryashov method. Finally, the conservation laws to the (3 + 1)-dimensional generalized BKP equation are presented by invoking the multiplier method.

Lie Symmetry Reductions and Wave Solutions of Coupled Equal Width Wave Equation

In the present paper, we have solved the coupled equal width wave equation by using the Lie symmetry analysis method. By applying the Lie symmetry transformation to the system of non-linear partial differential equations, we obtained the infinitesimal generators to the system and converted the system of non-linear partial differential equations into the system of non-linear ordinary differential equations. Further, the obtained system of ordinary differential equations has been solved by using the tanh method and power series method. The travelling wave solution to the coupled equal width wave equation has also been obtained. We have also found some implicit solution to the coupled equal width wave equation. The convergences of power series solutions have been shown.

The discrete tanh method for solving the nonlinear differential-difference equations

We investigated analytical solutions for the nonlinear differential-difference equations (DDEs) having fractional-order derivative. We employed the discrete tanh method in computations. Performance of trigonometric functions, dark one solitons and rational solutions are discussed in detail. The results with reliable parameters are illustrated via 2-D and 3-D graphs.

Computational solutions of the HIV-1 infection of CD4+T-cells fractional mathematical model that causes acquired immunodeficiency syndrome (AIDS) with the effect of antiviral drug therapy

This paper investigates the exact traveling wave solutions of the fractional model of the human immunodeficiency virus (HIV-1) infection for CD4T-cells. This model also treats with the effect of antiviral drug therapy. These solutions calculate both the boundary and initial conditions that allow employing the septic-B-spline scheme which is one of the most recent schemes in the numerical field. We use the obtained computational solutions via the modified Khater, the extended simplest equation, and sech–tanh methods through Atangana-Baleanu derivative operator. The comparison between the exact and numerical evaluated solutions is illustrated by some distinct sketches. The functioning of our numerical method is tested under three computational obtained solutions.

New interaction solutions and nonlocal symmetry of an equation combining the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form

The nonlocal symmetry is derived for an equation combining the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form from the truncated Painlevé expansion method. The nonlocal symmetries are localized to the Lie point symmetry by introducing new auxiliary dependent variables. The finite symmetry transformation and the Lie point symmetry for the prolonged system are solved directly. Many new interaction solutions among soliton and other types of interaction solutions for the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form can be obtained from the consistent condition of the consistent tanh expansion method by selecting the proper arbitrary constants.

Shock waves in magnetized electronegative plasma with nonextensive electrons

The nonlinear properties of ion acoustic shock waves in a magnetized electronegative plasma with nonextensive electrons and stationary dust grains are studied. The reductive perturbation technique is applied to derive the Zakharov–Kuznetsov Burgers (ZKB) equation. The Tanh method is used to obtain the shock wave solution. The present model is applied to four different electronegative plasma media which are found in space plasma, e.g. (H+ , H− ) plasma and (H+ , ) plasma, and laboratory experiments, e.g. (Xe+ , ) plasma and (Ar+ , ) plasma. It is found that the concentration of negative ions and the electron nonextensive parameter have important effects on the features of the propagated ion acoustic shock waves.

Phase plane analysis of the dust-acoustic waves for the Burgers equation in a strongly coupled dusty plasma

Phase plane analysis of dust-acoustic waves (DAWs) for the Burgers equation is investigated in a strongly coupled dusty plasma (SCDP) for the first time. The Burgers equation is obtained applying the reductive perturbation technique under the class of generalized hydrodynamic equations. The dynamical system of the Burgers equation is obtained using the traveling wave transfiguration. A phase plot containing a pair of heteroclinic orbits and a family of periodic orbits is presented for DAWs employing the concept of bifurcation theory through phase plane analysis. The analytical forms of kink and anti-kink wave solutions are derived by the tanh method. Analytical form of the periodic wave solution for the Burgers equation is derived for the first time in a SCDP. The parameters such as coupling parameter ( $$\Gamma$$ ), viscosity coefficient ( $$~\eta$$ ), dust–neutral collisional frequency ( $$\nu _{dn}$$ ), temperature of dusts ( $$T_d$$ ) and speed of wave (v) have significant effects on the dust-acoustic kink, anti-kink and periodic wave solutions.

A Note on the $$\exp (-\varphi (z))$$ Expansion Method

In this paper, we prove the equivalence between the $$\exp (-\varphi (z))$$ expansion method and the tanh method. The two methods give the same exact traveling wave solutions of nonlinear partial differential equations. We use the ill-posed Boussinesq equation as an example to demonstrate this fact.

Analytical modeling of electrical solitons in a nonlinear transmission line using Schamel–Korteweg deVries equation

Abstract Nonlinear transmission lines (NLTLs) provide an effective means of experimental investigation of nonlinear dispersive media paving the way for accurate modeling of electrical solitons. In this work, an appropriate analytical model is proposed to analyze and predict electrical solitons in an NLTL. The model is based on a new form of nonlinearity which utilizes fractional powers to render an accurate representation of the Capacitance-Voltage (C-V) curve, even for relatively higher voltages. A complete analytical procedure is adopted to obtain the soliton solution in an NLTL. Nonlinear analysis results in a Schamel type Korteweg deVries (S-KdV) equation, instead of a conventional KdV equation. An extended version of the Tanh method is employed to find the solution of S-KdV which yields narrower solitons of Sech4 shape. The proposed model is validated experimentally by using a specially designed NLTL (100 sections) comprising of linear inductors and nonlinear capacitances. Square pulses with a repetition rate of 100 kHz are used to generate the solitons. The comparative analysis shows excellent profile matching between theoretically predicted and experimentally obtained solitons. Furthermore, in consistence with the prediction of the proposed model, the product of the Amplitude and fourth power of the Width (A × W4) is found to be constant in experimental results. This model proposed in the present work can be of help in understanding and predicting the behavior of electrical solitons in NLTLs.

Highly dispersive optical solitons with cubic law and cubic-quintic-septic law nonlinearities

In this paper, the Kudryashov’s method and generalized tanh method (GTM) are employed to retrieve the new distinct optical solitons for highly dispersive NLSE. These two schemes effectively acquire the exact solution of the model in the form of dark, singular, singular periodic and bright-singular combo optical solitons. These solutions assist the mathematicians and physicians to realize the physical phenomena of present model. The solutions in presents work indicate that these schemes offer us an influential mathematical tools for solving NLSEs in various areas of applied sciences. In addition, the graphical illustrations of some solutions are also illustrated.

Lie symmetry analysis and traveling wave solutions of equal width wave equation

We obtained the power series solution and the traveling wave solutions of equal width wave equation by using the Lie symmetry method. The fundamental idea behind the symmetry transformation method is that it reduces one independent variables in a system of PDEs by utilizing Lie symmetries and surface invariance condition. We first obtained the infinitesimals and commutation table with the help of MAPLE software. Lie symmetry transformation method (STM) has been applied on EWW equation and converted it into various nonlinear ODEs. Then, the tanh method and the power series method have been applied for solving the reduced nonlinear ordinary differential equations (ODEs). Convergence of the power series solutions has also been shown.

Exact Solutions and Conservation Laws of the (3 + 1)-Dimensional B-Type Kadomstev–Petviashvili (BKP)-Boussinesq Equation

In this paper, Lie symmetry analysis is presented for the (3 + 1)-dimensional BKP-Boussinesq equation, which seriously affects the dispersion relation and the phase shift. To start with, we derive the Lie point symmetry and construct the optimal system of one-dimensional subalgebras. Moreover, according to the optimal system, similarity reductions are investigated and we obtain exact solutions of reduced equations by means of the Tanh method. In the end, we establish conservation laws using Ibragimov’s approach.

FUNCTIONAL EXPANSIONS FOR FINDING TRAVELING WAVE SOLUTIONS

The paper proposes a generalized analytic approach which allows to find traveling wave solutions for some nonlinear PDEs. The solutions are expressed as functional expansions of the known solutions of an auxiliary equation. The proposed formalism integrates classical approaches as tanh method or $G^{\prime }/G$ method, but it open the possibility of generating more complex solutions. A general class of second order PDEs is analyzed from the perspective of this formalism, and clear rules related to the balancing procedure are formulated. The KdV equation is used as a toy model to prove how the results obtained before through the $G^{\prime }/G$ approach can be recovered and extended, in an unified and very natural way.

Simultaneous feature weighting and parameter determination of Neural Networks using Ant Lion Optimization for the classification of breast cancer

In this paper, feature weighting is used to develop an effective computer-aided diagnosis system for breast cancer. Feature weighting is employed because it boosts the classification performance more as compared to feature subset selection. Specifically, a wrapper method utilizing the Ant Lion Optimization algorithm is presented that searches for best feature weights and parametric values of Multilayer Neural Network simultaneously. The selection of hidden neurons and backpropagation training algorithms are used as parameters of neural networks. The performance of the proposed approach is evaluated on three breast cancer datasets. The data is initially normalized using tanh method to remove the effects of dominant features and outliers. The results show that the proposed wrapper method has a better ability to attain higher accuracy as compared to the existing techniques. The obtained high classification performance validates the work which has the potential for becoming an alternative to the other well-known techniques.

Traveling wave solutions for the (3+1)-dimensional Davey-Stewartson equations

In this work, the extended tanh method is used to construct wave solutions for the Davey–Stewartson equations. The extended tanh method is a powerful solution method for obtaining different kind solutions of nonlinear evolution equations. This method can be applied to nonintegrable equations as well as to integrable ones.

On the traveling waves in nonlinear electrical transmission lines with intrinsic fractional-order using discrete tanh method

Abstract In this paper we investigate the solutions of the fractional partial differential-difference equations governing the dynamics of the voltage wave flowing inside two fractional (low pass and pass band) nonlinear electrical transmission lines (NETL). Through the discrete Tanh method, we derive the traveling wave solutions i.e (kink, dark, singular kink solitons) of two fractional partial differential-difference equations by using the fractional complex transform. The impact of the fractional order on the dynamical behavior of the analytical solutions in both models is highlighted. From our findings the perfect matching between the analytical and the numerical solutions is justified by the stability seen as the resulting wave propagates safely inside the fractional low pass NETL including real inductor. We get for the fractional pass band NETL a rise of nonlinear periodic waves and a train of multi periodic solitons during the numerical simulations. We show that, according to the modified Riemann–Liouville derivative properties, the findings can describe physical systems such as electrical systems and nonlinear electrical transmission lines with transient effect.

Optical Solitons With M-Truncated and Beta Derivatives in Nonlinear Optics

This paper studies optical solitons with M-truncated and beta derivatives (BD) for the Complex Ginzburg-Landau equation (CGLE) with Kerr Law nonlinearity. Two well-known integration schemes which are generalized tanh method (GTM) and generalized Bernoulli sub-ODE method (GBM) are utilized to extract such optical soliton solutions. For the successful existence of the solutions, the constraints conditions have been presented. The discussion for the physical features of the obtained solutions is reported.