Search

We adapt Sklyanin’s K-matrix formalism to the sinh–Gordon equation, and prove that all free boundary constant mean curvature annuli in the unit ball in R 3 are of finite type.

The polynomials used in the formation of the probability distribution density function of the hyperbolic cosine type are investigated. Earlier, on the basis of a hyperbolic cosine distribution, the author obtained numerical sets, among which not only new ones, but also, for example, the triangle of Stirling numbers, the triangle of the coefficients of Bessel polynomials, sequences of coefficients in the expansion of various functions, etc. In this paper, depending on the natural parameter m and the real distribution parameter β , a new class of polynomials is obtained. For even and odd m , the polynomials are constructed using similar, but different formulas. The article presents polynomials for even values m . Structurally, polynomials consist of quadratic factors. The coefficients of the polynomials, ordered by m , form numerical triangles depending on β . Some relations are found between the coefficients. From the numerical triangles, a set of numerical sequences is obtained, which for integers β are integers. Also, polynomials with respect to x turn out to be polynomials with respect to β . With this interpretation the variable acts as a parameter. New numerical triangles and sequences for different x were found. The overwhelming majority of the obtained numerical sequences are new. The class of polynomials arising from problems of probability theory indicates the possibility of applying the results.

The consistent tanh expansion (CTE) method is applied to the (2+1)-dimensional Boussinesq equation which describes the propagation of ultrashort pulse in quadratic nonlinear medium. The interaction solutions are explicitly given, such as the bright soliton-periodic wave interaction solution, variational amplitude periodic wave solution, and kink-periodic wave interaction solution. We also obtain the bright soliton solution, kind bright soliton solution, double well dark soliton solution and kink-bright soliton interaction solution by using Painleve truncated expansion method. And we investigate interactive properties of solitons and periodic waves.

The present study compares the performance of two rainfall-runoff models including Tanh model and AWBM model for streamflow simulation. The Tanh model is a 2-parameter empirical model, and the AWBM is a 9-parameter conceptual model. The Tullaroop Creek sub-catchment at central Victoria is selected as the study area. Rainfall and streamflow data measured from the catchment are the major inputs to these models. The Tanh function is used as a simple rainfall-runoff relationship for the Tullaroop Creek catchment and it estimates annual runoff by using curve fitting of the rainfall runoff plot. The AWBM model is calibrated and validated for two periods: 1974-1987 and 1987-2000, respectively. The Nash–Sutcliffe Efficiency is computed to evaluate the efficacy of model predictions. From the results obtained, it is found that the AWBM model can provide adequate estimates of annual runoff at the study catchments than Tanh model.

A Ground-penetrating radar (GPR) is considered an efficient non-destructive device for detecting the buried object. The GPR operation is based on the analysis of the received scattered of the transmitted signal. Its output is the two-dimensional image radiogram hyperbolic curve, this image represents the reflected signal of the buried object. To date, studies investigating targets (buried objects) have produced equivocal results. Therefore, this paper highlights the importance of the effect of changing target-detection parameters (material type, target to surface distance, size, and scan to target location angle) concerning the hyperbolic curve on the GPR experiment image. A practical model has been built for this experiment with three material types (metal, plastic, and pottery) that are buried in the sand soil. Three tests have been done for different types of material, in the different depth for each buried object of the same material. Then, changing the size of the buried object (small and big size), also, changing the location angle of the buried objects concerning the direction of the GPR scan. For these tests, the effect on the hyperbolic curve has been recognized. A MALA 1 GHz geophysical GPR system is used in these experiments. As a result, the strength of the reflected EMW changes concerning the type of material, size, depth, and location angle of the buried object.

Through investigating the sinh-Gaussian beam with complex beam parameters, it is the first time to find that such beams can carry vortices and exhibit dark hollow intensity distributions when the complex beam parameters are sufficiently small. A closed-form propagation equation for sinh-Gaussian beams through paraxial ABCD optical systems is derived based on the Collins formula and illustrated with numerical methods. It is shown that the perfect hollow configuration can retain a quite long propagating distance under small complex beam parameters. The analytical discussions affirm the numerical conclusions.

An exact representation of the Baker–Campbell–Hausdorff formula as a power series in just one of the two variables is constructed. Closed form coefficients of this series are found in terms of hyperbolic functions, which contain all of the dependence on the second variable. It is argued that this exact series may then be truncated and be expected to give a good approximation to the full expansion if only the perturbative variable is small. This improves upon existing formulae, which require both to be small. Several different representations are provided and emphasis is given to the situation where one of the matrices is diagonal, where a particularly easy to use formula is obtained.

The pedestal characteristic is an important basis for high confinement mode (H-mode) research. Because of the finite spatial resolution of Thomson scattering (TS) diagnostic on Experimental Advanced Superconducting Tokamak (EAST), it is necessary to characterize the pedestal with a suitable functional form. Based on simulated and experimental data of EAST, it is shown that the two-line method with a bilinear fitting has better reproducibility of pedestal parameters than hyperbolic tangent (tanh) and modified hyperbolic tangent (mtanh) methods. This method has been applied to EAST type I edge localized mode (ELM) discharges, and the electron pedestal density is found to be proportional to the line-averaged density and the edge pressure gradient is found to be proportional to the pedestal pressure. Furthermore, the ion poloidal gyro-radius has been identified as the suitable parameter to describe the pedestal pressure width.

The consistent tanh expansion (CTE) method is employed to the (2+1)-dimensional Caudrey—Dodd—Gibbon-Kotera—Sawada (CDGKS) equation. The interaction solutions between solitons and the cnoidal periodic waves are explicitly obtained. Concretely, we discuss a special kind of interaction solution in the form of tanh functions and Jacobian elliptic functions in both analytical and graphical ways. The results show that the profiles of the soliton-cnoidal periodic wave interaction solutions can be designed by choosing different values of wave parameters.

A consistent tanh expansion (CTE) method is used to study the modified Boussinesq equation. It is proved that the modified Boussinesq equation is CTE solvable. The soliton-cnoidal periodic wave is explicitly given by a nonanto-BT theorem. Furthermore, the nonlocal symmetry for the modified Boussinesq equation is obtained by the Painleve analysis. The nonlocal symmetry is localized to the Lie point symmetry by introducing one auxiliary dependent variable. The finite symmetry transformation related with the nonlocal symemtry is obtained by solving the initial value problem of the prolonged systems. Thanks to the localization process, many interaction solutions among solitons and other complicated waves are computed through similarity reductions. Some special concrete soliton-cnoidal wave interaction behaviors are studied both in analytical and graphical ways.

Many longitudinal studies have collected serial body core temperature (T c) data to understand thermal work strain of workers under various environmental and operational heat stress environments. This provides the opportunity for the development of mathematical models to analyse and forecast temporal T c changes across populations of subjects. Such models can reduce the need for invasive methods that continuously measure T c. This current work sought to develop a nonlinear mixed effects modelling framework to delineate the dynamic changes of T c and its association with a set of covariates of interest (e.g. heart rate, chest skin temperature), and the structure of the variability of T c in various longitudinal studies. Data to train and evaluate the model were derived from two laboratory investigations involving male soldiers who participated in either a 12 (N = 18) or 15 km (N = 16) foot march with varied clothing, load and heat acclimatisation status. Model qualification was conducted using nonparametric bootstrap and cross validation procedures. For cross validation, the trajectory of a new subject's T c was simulated via Bayesian maximum a posteriori estimation when using only the baseline T c or using the baseline T c as well as measured T c at the end of every work (march) phase. The final model described T c versus time profiles using a parametric function with its main parameters modelled as a sigmoid hyperbolic function of the load and/or chest skin temperature. Overall, T c predictions corresponded well with the measured data (root mean square deviation: 0.16 °C), and compared favourably with those provided by two recently published Kalman filter models.

The nonlocal symmetry for the potential Kadomtsev–Petviashvili (pKP) equation is derived by the truncated Painleve analysis. The nonlocal symmetry is localized to the Lie point symmetry by introducing the auxiliary dependent variable. Thanks to localization process, the finite symmetry transformations related with the nonlocal symmetry are obtained by solving the prolonged systems. The inelastic interactions among the multiple-front waves of the pKP equation are generated from the finite symmetry transformations. Based on the consistent tanh expansion method, a nonauto-Backlund transformation (BT) theorem of the pKP equation is constructed. We can get many new types of interaction solutions because of the existence of an arbitrary function in the nonauto-BT theorem. Some special interaction solutions are investigated both in analytical and graphical ways.

In this paper, we propose an innovative chaotic system, combining fractional derivative and sine-hyperbolic nonlinearity with circuit execution. The study of this system is conducted via an analog circuit simulator, using two anti-parallel semiconductor diodes to provide hyperbolic sine nonlinearity, and to function as operational amplifiers. The multi-stability of the system is also enhanced with the help of multi-equilibrium points for distinct real orders of system. The system explores the generation of a four-wing attractor in different phases, both numerically and electronically. By changing the input parameters of the system, different graphs are generated for current flow in state, phase, and space, to confirm the precision of the fractional order derivatives. A reasonable simulation shows that the deliberate circuit provides effective chaos in terms of speed and accuracy, which is comensurate with the numerical simulation. This nonlinear chaotic behavior is utilized to encrypt sound (.wav), images (.jpg), and animated (.gif) data which are a major requirement for the security of communication systems. The proposed circuit performs chaos and cryptographic tasks with high-effective analog computation, and constitutes a novel approach to this area of research.

Improvements are proposed in this paper to overcome the drawbacks of traditional fuzzy enhancement algorithms. Tanh operator is used as membership function, and fuzzy enhancement function are made up of simple power functions, and the image is divided into two regions by adaptive segmentation method, one is high grey region, the other is low grey region, pixels in one region are enhanced, and pixels in the other region are reduced. Simulation results show that this algorithm has good ability to increase image’s contrast, and it is a simple and efficient way to enhance blur edges.

The comparison of results obtained using the artificial neural networks (ANN) has been carried out in researching solar radiation for various regions of the Russian Federation with data modeling applying a spline interpolation preserving an interpolant shape or a polynomial (up to tenth degree) in the MATLAB environment obtained when using the MATLAB Basic Fitting-a graphical user interface (GUI). The neural network has been trained by the feed-forward backprop algorithm. The following functions were used: the Bayesian Regularization, the function of gradient descent with regard for moments, the hyperbolic tangent functions. The Mean Square Error (MSE) was chosen as a loss function which was minimized. The 15 input parameters were taken. As a result, the obtained model had high values of correlation coefficients and low values of the Mean Square Error among the target and the output values. The polynomial (up to tenth degree inclusive) coefficients and the norm of the residuals were obtained which were decreasing with the increasing of the polynomial degree. It was shown that the better results are received using the artificial neural networks (ANN), somewhat worse ones employing a spline interpolant and shape-preserving interpolant. At the same time, the proposed models and methods can be used in calculation of solar radiation for various regions in the Russian Federation and other countries.

The aim of this work is to regularize the expression of the coefficients, expressed in term of trigonometrical or hyperbolic functions, arising from the exponential fitting procedure, by reformulating them in terms of the so-called ηm functions. These coefficients are functions of the variable ν = ωh where ω is the frequency and h is the step size. This reformulation eliminates the 0/0 indeterminate form of the coefficients when ν tends to 0. This procedure makes the methods more accurate. A numerical evidence is also given.

Firstly, the parameters of hyperbolic tangent model with magneto-rheological (MR) damper are identified via particle swarm optimization algorithm based on the mechanical properties test. Then, the functional relationship between the identified parameters and input current is fitted by curve fitting toolbox. Ultimately, Simulink toolbox is utilized to design and establish the hyperbolic tangent simulation model with MR damper, meanwhile, the different input currents and sinusoidal signals with other amplitudes and frequencies are selected for simulation and comparative analysis. The results show that parameter identification accuracy of particle swarm optimization algorithm is high. The simulation and experiment data under combination conditions with different input currents and other sinusoidal signals are in good agreement, which verifies the generality and accuracy of the parameter identification results.

In this study, a novel parametric model for magnetorheological (MR) dampers considering excitation characteristics (amplitude and frequency) is proposed. A MR damper is tested under different working conditions by adjusting the input current and changing the excitation amplitude and frequency, and the damping characteristics of the MR damper are obtained. And then a parametric model (Tanh model) describing the nonlinear dynamics of MR dampers is established and the parameters are fitted based on the test results. Further, a modified model considering the excitation characteristics is proposed, and the capability and accuracy of the two models in modeling and predicting the damping characteristics of the MR damper are compared in the case of changing the input current, excitation amplitude and excitation frequency. The results show that the proposed model has higher precision than the Tanh model, which can adapt well to the changes of current, excitation amplitude and excitation frequency, and can accurately predict the damping characteristics under the working conditions outside the test range. In addition, the proposed model is easy to invert, so a feed-forward control can be designed based on the obtained inverse model to realize the damping force tracking control of MR dampers.

In this paper, we used the modied auxiliary expansion method to nd some new family solution of the paraxial nonlinear Schrodinger equation. The solutions have a hyperbolic function, trigonometric function, exponential function, and rational function forms. The linear stability analysis of paraxial NLSE is also studied. Two cases when the instability modulation becomes to occur are investigated. All solutions are new and veried the main equation of the paraxial wave equation. Moreover, the constraint conditions for the existence of soliton solutions are also showed.

Using the standard truncated Painlevé expansion, the residual symmetry of the (2+1)-dimensional modified dispersive water-wave system is localized in the properly prolonged system with the Lie point symmetry vector. Some different transformation invariances are derived by utilizing the obtained symmetries. The symmetries of the system are also derived through the Clarkson–Kruskal direct method, and several types of explicit reduction solutions relate to the trigonometric or the hyperbolic functions are obtained. Finally, some special solitons are depicted from one of the solutions.

In this paper, the Clarkson-Kruskal direct approach is employed to investigate the exact solutions of the 2-dimensional rotational Euler equations for the incompressible fluid. The application of the method leads to a system of completely solvable ordinary differential equations. Several special cases are discussed and novel nonlinear exact solutions with respect to variables x and y are obtained. It is of interest to notice that the pressure p is obtained by the second kind of curvilinear integral and the coefficients of the nonlinear solutions are solitary wave type functions like tanh(kt/2) and sech (kt/2) due to the rotational parameter k ≠ 0. Such phenomenon never appear in the classical Euler equations wherein the Coriolis force arising from the gravity and Earth's rotation is ignored. Finally, illustrative numerical figures are attached to show the behaviors that the exact solutions may exhibit.

In this paper, the truncated Painlevé analysis and the consistent tanh expansion (CTE) method are developed for the (2+1)-dimensional breaking soliton equation. As a result, the soliton-cnoidal wave interaction solution of the equation is explicitly given, which is difficult to be found by other traditional methods. When the value of the Jacobi elliptic function modulus m = 1, the soliton-cnoidal wave interaction solution reduces back to the two-soliton solution. The method can also be extended to other types of nonlinear evolution equations in mathematical physics.

The nonlocal symmetries for the higher-order KdV equation are obtained with the truncated Painleve method. The nonlocal symmetries can be localized to the Lie point symmetries by introducing suitable prolonged systems. The finite symmetry transformations and similarity reductions for the prolonged systems are computed. Moreover, the consistent tanh expansion (CTE) method is applied to the higher-order KdV equation. These methods lead to some novel exact solutions of the higher-order KdV system.

In this letter, we investigate traveling wave solutions of a nonlinear wave equation with degenerate dispersion. The phase portraits of corresponding traveling wave system are given under different parametric conditions. Some periodic wave and smooth solitary wave solutions of the equation are obtained. Moreover, we find some new hyperbolic function compactons instead of well-known trigonometric function compactons by analyzing nilpotent points.

A consistent tanh expansion (CTE) method is developed for the dispersion water wave (DWW) system. For the CTE solvable DWW system, there are two branches related to tanh expansion, the main branch is consistent while the auxiliary branch is not consistent. From the consistent branch, we can obtain infinitely many exact significant solutions including the soliton-resonant solutions and soliton-periodic wave interactions. From the inconsistent branch, only one special solution can be found. The CTE related nonlocal symmetries are also proposed. The nonlocal symmetries can be localized to find finite Backlund transformations by prolonging the model to an enlarged one.

A Laplace decomposition algorithm is adopted to investigate numerical solutions of a class of nonlinear partial differential equations with nonlinear term of any order, utt + auxx + bu + cup + du2p−1 = 0, which contains some important equations of mathematical physics. Three distinct initial conditions are constructed and generalized numerical solutions are thereby obtained, including numerical hyperbolic function solutions and doubly periodic ones. Illustrative figures and comparisons between the numerical and exact solutions with different values of p are used to test the efficiency of the proposed method, which shows good results are achieved.

A three-parameter probability distribution is presented, modeled on the basis of the negative binomial distribution. This distribution corresponds structurally to the author’s distribution of the hyperbolic cosine type by the form of the characteristic function. The difference lies in the use of standard trigonometric cosine and sine for the characteristic function instead of the corresponding hyperbolic functions. The moment-forming polynomials of two arguments are found, derived from the recurrent differential relation to calculate the moments of distribution. A recurrent algebraic formula to produce the integer coefficients of these polynomials is introduced. The set of the coefficients depending on three arguments is ordered and geometrically interpreted as a numerical prism. Numerical prism sections are numerical triangles and numerical sequences. Among the sections of the prism there are well-known and new numerical sets, for example, Stirling numerical triangle, Bessel numerical triangle with alternating signs of elements, alternating tangential numbers, etc. A connection with the numerical prism derived from the hyperbolic cosine type distribution is indicated.

in the present article, we present the peristaltic motion of “Hyperbolic Tangent nanofluid” by a porous area in a two dimensional non-regular a symmetric channel with an inclination under the impact of inclination angle under the impact of inclined magnetic force, the convection conditions of “heat and mass transfer” will be showed. The matter of the paper will be further simplified with the assumptions of long wave length and less “Reynolds number”. we are solved the coupled non-linear equations by using technical analysis of “Regular perturbation method” of series solutions. We are worked out the basic equations of continuity, motion, temperature, and volume fraction particles for the recently fluid. The impact of incoming parameters on the inflow features have been studied and painted.

Solitonic solutions have become a popular topic among traveling wave solutions since they act as a bridge between mathematics and physics. In this article, two newly developed methods, namely the extended rational sine-cosine method and extended rational sinh-cosh method by means of fractional complex transform are applied to construct the exact solutions in the form of hyperbolic, trigonometric and rational functions of the nonlinear fractional Phi-4 equation. The obtained solutions are further expressed by solitons, dark solitons, periodic and kink wave solutions. The proposed methods analyze the possibility of the existence of limit cycles with specific parameters.

The time-dependent boundary element method (BEM) is widely used to solve transient heat conduction problem. The integrands of the domain integrals in the time-dependent BEM are close to singular when small time step is used. A straightforward computation of these integrals using Gaussian quadrature will produce large errors. To evaluate domain integrals accurately and efficiently, the sinh transformation combined with element subdivision method is proposed in this paper. With the sinh transformation, the integrands of the domain integrals become smoother. Meanwhile, the Gaussian points are shifted towards the source point due to the element subdivision. Thus, the 2D domain integrals in the time-dependent BEM can be computed accurately. Numerical examples have demonstrated the accuracy and efficiency of the proposed method.

This paper proposes support vector machines (SVMs), which is currently one of the most popular algorithms in machine learning (ML), in order to classify the low birth weight (LBW) data. The main objectives of this study are to predict the classification of LBW data in Indonesia based on the SVMs andto compare the performance of the proposed SVMs with the binary logistic regression as the most common model for classification of LBW data. The obtained samples were based on the results of Indonesian Demographic and Health Survey in 2012. The results showed that SVMs with four kernel functions (linear, radial, polynomial and hyperbolic tangent) were fit well to the LBW data in Indonesia. Furthermore, the constructed SVMs based on linear kernel function had the best performance among the SVMs with the other proposed kernel functions. This research also concluded that the SVMs based on linear kernel competed well with thebinary logistic regression forclassification LBW data in Indonesia.

Through this article, we studied the peristaltic motion of “Hyperbolic Tangent” fluid in the geometry of curvature channel by using the analysis of large wavelength and less of Reynolds number. The matter has controlled mathematically by the partial differential equations of continuity, motion, heat transfer. In the study, we used the impact of radial magnetic force. The obtained coupled non-linear equations of above equations have solved by an approximation technical. Locked formula solutions of the stream function, axial velocity, heat function has evaluated. The influence of curvature is analysed and took it into account. The impact of sundry variables on the inflow features have plotted and explained by graphs and figures.

Some exact solutions of KdV-Burgers-Kuramoto (KBK) equation are derived by the anzas and tanh methods. Also, the most general Lie point symmetry group of the KBK equation are presented using the basic Lie symmetry method. As well as, the non-classical and weak symmetries of this equation, as well as the corresponding similarity reductions, are investigated. Finally, the classical and non-classical symmetries of KBK and KdV-Burgers (KB) equations are compared.

In order to weaken the chattering of the sliding mode control and improve the convergence speed of the approaching motion phase and the sliding mode motion phase.Introducing the inverse tangent function of the absolute value of the state variable and the hyperbolic sine function of the switching function to design a new reaching law,combined with non-singular fast terminal sliding mode to design control law.In order to reduce the effects of viscosity coefficient and load torque disturbance,and improve the robustness of the permanent magnet synchronous motor control system.Observing the disturbance with a second-order extended state observer,the observations are introduced into the sliding mode speed controller for feedforward compensation.The nonlinear function adopts hyperbolic sine function to avoid the observer parameter being too large.The simulation result shows that second-order extended state observer is ability to quickly and accurately track disturbances.Compared to PI control,sliding mode control with feedforward compensation, no overshoot, faster response and stronger robustness.

The Boussinesq equation is one of important prototypic models in nonlinear physics. Various nonlinear excitations of the Boussinesq equation have been found by many methods. However, it is very difficult to find interaction solutions among different types of nonlinear excitations. In this peper, two equivalent very simple methods, the truncated Painleve analysis and the generalized tanh function expansion approaches, are developed to find interaction solutions between solitons and any other types of Boussinesq waves.

In the present paper, we have investigated the peristaltic flow of hyperbolic tangent fluid in a curved channel. The governing equations of hyperbolic tangent fluid model for curved channel are derived including the effects of curvature. The highly nonlinear partial differential equations are simplified by using the wave frame transformation, long wave length and low Reynolds number assumptions. The reduced nonlinear partial differential equation is solved analytically with the help of homotopy perturbation method (HPM). The physical features of pertinent parameters have been discussed by plotting the graphs of pressure rise and stream functions.

In this paper, an extended Riccati sub-ODE method is proposed to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann—Liouville derivative. By a fractional complex transformation, a given fractional differential-difference equation can be turned into another differential-difference equation of integer order. The validity of the method is illustrated by applying it to solve the fractional Hybrid lattice equation and the fractional relativistic Toda lattice system. As a result, some new exact solutions including hyperbolic function solutions, trigonometric function solutions and rational solutions are established.

International Conference on Quantum Science and Applications, ICQSA 2016; Eskisehir Osmangazi University Congress and Culture CentreEskisehir; Turkey; 25 May 2016 through 27 May 2016, We extend the concept of q-analytic function in two different directions. First we find expansion of q-binomial in terms of q-Hermite polynomials, analytic in two complex arguments. Based on this representation, we introduce a new class of complex functions of two complex arguments, which we call the double q-analytic functions. As another direction, by the hyperbolic version of q-analytic functions we describe the q-analogue of traveling waves, which is not preserving the shape during evolution. The IVP for corresponding q-wave equation we solved in the q-D'Alembert form.

Using the machinery of Lie group analysis, the nonlinear system of coupled Burgers-type equations is studied. Using the infinitesimal generators in the optimal system of subalgebra of the said Lie algebras, it leads to two nonequivalent similarity transformations by using it we obtain two reductions in the form of system of nonlinear ordinary differential equations. The search for solutions of these systems by using the G'/G-method has yielded certain exact solutions expressed by rational functions, hyperbolic functions, and trigonometric functions. Some figures are given to show the properties of the solutions.

First, two tanh-coth type solutions of a class of nonlinear wave equation are derived by using a simplified modified extended tanh-function method. Then, further analysis to some tanh-coth type solutions of nonlinear evolution equations are given. The results show that when balance number m is one or two, the tanh-coth type solutions obtained by the modified extended tanh-function method can be obtained by using the hyperbolic-function method.

Based on a transformed Painleve property and the variable separated ODE method, a function transformation method is proposed to search for exact solutions of some partial differential equations (PDEs) with hyperbolic or exponential functions. This approach provides a more systematical and convenient handling of the solution process of this kind of nonlinear equations. Its key point is to eradicate the hyperbolic or exponential terms by a transformed Painleve property and reduce the given PDEs to a variable-coefficient ordinary differential equations, then we seek for solutions to the resulting equations by some methods. As an application, exact solutions for the combined sinh-cosh-Gordon equation are formally derived.

New efficient analytic approximations have been found for the modified zero-order Bessel functions I 0(x). The method, used herein, improves prior techniques, such as multipoint quasi-rational approximations, MPQA. The present work is also an improvement of previous works, in the sense that the form of the approach is more efficient, that is, a smaller relative error is obtained using a lower number of parameters. As in the Pade method rational functions are used with coefficients determined from the powers series, but now also asymptotic expansions are also used simultaneously with that series, and consequently the rational functions have to be combined with elementary functions, in such a way, that the approach is a bridge between both expansions. The approximation now found is a rational function combined with a hyperbolic function. Only three parameters are needed to obtain a relative error smaller than 0.6 percent.

Abundant new exact solutions of the Schamel–Korteweg-de Vries (S-KdV) equation and modified Zakharov–Kuznetsov equation arising in plasma and dust plasma are presented by using the extended mapping method and the availability of symbolic computation. These solutions include the Jacobi elliptic function solutions, hyperbolic function solutions, rational solutions, and periodic wave solutions. In the limiting cases, the solitary wave solutions are obtained and some known solutions are also recovered.