The Modified Simple Equation Method and the Multiple Exp-function Method for Solving Nonlinear Fractional Sharma-Tasso-Olver Equation

In this article, the fractional derivatives in the sense of the modified Riemann-Liouville derivatives together with the modified simple equation method and the multiple exp-function method are employed for constructing the exact solutions and the solitary wave solutions for the nonlinear time fractional Sharma-Tasso- Olver equation. With help of Maple, we can get exact explicit l-wave, 2-wave and 3-wave solutions, which include l-soliton, 2-soliton and 3-soliton type solutions if we use the multiple exp-function method while we can get only exact explicit l-wave solution including l-soliton type solution if we use the modified simple equation method. Two cases with specific values of the involved parameters are plotted for each 2-wave and 3-wave solutions.

Exact Solutions of Kolmogorov-Petrovskii-Piskunov Equation Using the Modified Simple Equation Method

The modified simple equation method is employed to find the exact solutions of the nonlinear Kolmogorov-Petrovskii-Piskunov （KPP） equation. When certain parameters of the equations are chosen to be special values, the solitary wave solutions are derived from the exact solutions. It is shown that the modified simple equation method provides an effective and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

Solitary Wave Solutions to the ZKBBM Equation and the KPBBM Equation via the Modified Simple Equation Method

In this article, the modified simple equation method （MSE） is used to acquire exact solutions to nonlinear evolution equations （NLEEs） namely the Zakharov- Kuznetsov Benjamin-Bona-Mahony equation and the Kadomtsov-Petviashvilli Benjamin-Bona-Mahony equation which have widespread usage in modern science. The MSE method is ascending and useful mathematical tool for constructing exact travel- ing wave solutions to NLEEs in the field of science and engineering. By means of this method we attained some significant solutions with free parameters and for special values of these parameters, we found some soliton solutions derived from the exact solutions. The solutions obtained in this article have been shown graphically and also discussed physically.

EXACT SOLUTIONS TO KdV-BURGERS EQUATION USING THE ENHANCED MODIFIED SIMPLE EQUATION METHOD

It is known that many physical phenomenons can be described by the KdV-Burgers equation. By the enhanced modifed simple equation method, we obtain the exact solutions with variable coefcients involving parameters to KdV-Burgers equation, and some new exact solutions are gained. When these parameters are taken special values, the solitary wave solutions are derived from the exact solutions. It is shown that the proposed method provides a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

Propagation of traveling wave solutions to the Vakhnenko-Parkes dynamical equation via modified mathematical methods

In this paper, we investigate some new traveling wave solutions to Vakhnenko-Parkes equation via three modified mathematical methods. The derived solutions have been obtained including periodic and solitons solutions in the form of trigonometric, hyperbolic, and rational function solutions. The graphical representations of some solutions by assigning particular values to the parameters under prescribed conditions in each solutions and comparing of solutions with those gained by other authors indicate that these employed techniques are more effective, efficient and applicable mathematical tools for solving nonlinear problems in applied science.

Stem taper equations for poplars growing on farmland in Sweden

We developed a simple polynomial taper equation for poplars growing on former farmland in Sweden and also evaluated the performance of some well-known taper equations. In Sweden there is an increasing interest in the use of poplar. Effective management of poplar plantations for high yield production would be facilitated by taper equations providing better predictions of stem volume than currently available equations. In the study a polynomial stem taper equation with five parameters was established for individual poplar trees growing on former farmland. The outputs of the polynomial taper equation were compared with five published equations. Data for fitting the equations were collected from 69 poplar trees growing at 37 stands in central and southern Sweden （lat. 55–60° N）. The mean age of the stands was 21 years （range 14–43）, the mean density 984 stems·ha？1 （198–3,493）, and the mean diameter at breast height （outside bark） 25 cm （range 12–40）. To verify the tested equations, performance of accuracy and precision diameter predictions at seven points along the stem was closely analyzed. Statistics used for evaluation of the equations indicated that the variable exponent taper equation presented by Kozak （1988） performed best and can be recommended. The stem taper equation by Kozak （1988） recommended in the study is likely to be beneficial for optimising the efficiency and profitability of poplar plantation management. The constructed polynomial equation and the segmented equation presented by Max ＆ Burkhart （1976） were second and third ranked. Due to the statistical complexity of Kozak’s equation, the constructed polynomial equation is alternatively recommended when a simple model is requested and larger bias is accepted.

An unsymmetric constitutive equation for anisotropic viscoelastic fluid

A continuum constitutive theory of corotational derivative type is developed for the anisotropic viscoelastic fluid-liquid crystalline （LC） polymers. A concept of anisotropic viscoelastic simple fluid is introduced. The stress tensor instead of the velocity gradient tensor D in the classic Leslie-Ericksen theory is described by the first Rivlin-Ericksen tensor A and a spin tensor W measured with respect to a co-rotational coordinate system. A model LCP-H on this theory is proposed and the characteristic unsymmetric behaviour of the shear stress is predicted for LC polymer liquids. Two shear stresses thereby in shear flow of LC polymer liquids lead to internal vortex flow and rotational flow. The conclusion could be of theoretical meaning for the modern liquid crystalline display technology. By using the equation, extrusion-extensional flows of the fluid are studied for fiber spinning of LC polymer melts, the elongational viscosity vs. extension rate with variation of shear rate is given in figures. A considerable increase of elongational viscosity and bifurcation behaviour are observed when the orientational motion of the director vector is considered. The contraction of extru- date of LC polymer melts is caused by the high elongational viscosity. For anisotropic viscoelastic fluids, an important advance has been made in the investigation on the constitutive equation on the basis of which a seriesof new anisotropic non-Newtonian fluid problems can be addressed.

Comparative numerical study of single and two-phase models of nanofluid heat transfer in wavy channel

The main purpose of this study is to survey numerically comparison of two- phase and single phase of heat transfer and flow field of copper-water nanofluid in a wavy channel. The computational fluid dynamics （CFD） prediction is used for heat transfer and flow prediction of the single phase and three different two-phase models （mixture, volume of fluid （VOF）, and Eulerian）. The heat transfer coefficient, temperature, and velocity distributions are investigated. The results show that the differences between the temperature fie].d in the single phase and two-phase models are greater than those in the hydrodynamic tleld. Also, it is found that the heat transfer coefficient predicted by the single phase model is enhanced by increasing the volume fraction of nanoparticles for all Reynolds numbers; while for the two-phase models, when the Reynolds number is low, increasing the volume fraction of nanoparticles will enhance the heat transfer coefficient in the front and the middle of the wavy channel, but gradually decrease along the wavy channel.

Existence of Exactly n+1 Simple 4-Periodic Solutions of the Differential Delay Equation x(t)=-f(x(t-1))

By discussing the zeros of periodic.solutions we give in this paper a criterion for the existence of exactly n+1 simple 4-periodic solutions of the differential delay equation x(T)= -f(x(t-1)).

Densities and Viscosities of the Quaternary System Mannitol-Sorbitol- D-Glucose-H2O and Its Ternary Subsystems at 298.15K

To check the applicabilities of the simple density equation and viscosity equation in the semi-ideal solution theory to nonelectrolyte solutions, the densities and viscosities were measured for the quaternary system mannitol-sorbitol-D-glucose-HzO and its ternary subsystems mannitol-D-glucose-H2O and sorbitol-D-glucose-H2O at 298.15K. The results were used to test the applicability of the simple equations for the density and viscosity of the multicomponent nonelectrolyte solution. The agreements between the predicted and measured results are good.

Recurrent formula of Bernoulli numbers and the relationships among the coefficients of beam,Bernoulli numbers and Euler numbers

Based on the differential equation of the deflection curve for the beam,the equation of the deflection curve for the simple beamis obtained by integral. The equation of the deflection curve for the simple beamcarrying the linear load is generalized,and then it is expanded into the corresponding Fourier series.With the obtained summation results of the infinite series,it is found that they are related to Bernoulli num-bers and π. The recurrent formula of Bernoulli numbers is presented. The relationships among the coefficients of the beam,Bernoulli numbers and Euler numbers are found,and the relative mathematical formulas are presented.

Dispersive soliton solutions for shallow water wave system and modified Benjamin-Bona-Mahony equations via applications of mathematical methods

We have utilized three novel methods,called generalized direct algebraic,modified F-expansion and improved simple equation methods to construct traveling wave solutions of the system of shallow water wave equations and modified Benjamin-Bona-Mahony equation.After substituting particular values of the parameters,different solitary wave solutions are derived from the exact traveling wave solutions.It is shown that these employed methods are more powerful tools for nonlinear wave equations.

Bright, periodic, compacton and bell-shape soliton solutions of the extended QZK and (3+1)-dimensional ZK equations

The(3+1)-dimensional Zakharov–Kuznetsov(ZK) and the new extended quantum ZK equations are functional to decipher the dense quantum plasma, ion-acoustic waves, electron thermal energy,ion plasma, quantum acoustic waves, and quantum Langmuir waves. The enhanced modified simple equation(EMSE) method is a substantial approach to determine competent solutions and in this article, we have constructed standard, illustrative, rich structured and further comprehensive soliton solutions via this method. The solutions are ascertained as the integration of exponential, hyperbolic,trigonometric and rational functions and formulate the bright solitons, periodic, compacton, bellshape, parabolic shape, singular periodic, plane shape and some new type of solitons. It is worth noting that the wave profile varies as the physical and subsidiary parameters change. The standard and advanced soliton solutions may be useful to assist in describing the physical phenomena previously mentioned. To open out the inward structure of the tangible incidents, we have portrayed the three-dimensional, contour plot, and two-dimensional graphs for different parametric values. The attained results demonstrate the EMSE technique for extracting soliton solutions to nonlinear evolution equations is efficient, compatible and reliable in nonlinear science and engineering.

New conservation laws and exact solutions of the special case of the fifth-order KdV equation

The current study deals with the Kaup-Kupershmidt(KK)equation to construct formal Lagrangian,con-servation laws,and exact solutions.KK is basically a special case of the 5th-order KdV equation.The conservation laws obtained by using the conservation theorem are trivial conservation laws.In addition,exact solutions are found via the modified simple equation(MSE)method.For a suitable value of solu-tions,the 3D surfaces have been plotted using MAPLE.These plots giving novel exact solutions are made to reveal important wave characteristics.Our obtained results in this work concerning our investigated equation are essential to explain many physical and oceanographic applications involving ocean gravity waves and many other related phenomena.

N1-soliton solution for Schrodinger equation with competing weakly nonlocal and parabolic law nonlinearities

The nonlocal nonlinear Schr?dinger equation(NNLSE)with competing weakly nonlocal nonlinearity and parabolic law nonlinearity is explored in the current work.A powerful integration tool,which is a modified form of the simple equation method,is used to construct the dark and singular 1-soliton solutions.It is shown that the modified simple equation method provides an effective and powerful mathematical gadget for solving various types of NNLSEs.

On Solving the （2＋1）-Dimensional Nonlinear Cubic-Quintic Ginzburg-Landau Equation Using Five Different Techniques

In this article, we apply five different techniques, namely the （G//G）-expansion method, an auxiliary equation method, the modified simple equation method, the first integral method and the Riccati equation method for constructing many new exact solutions with parameters as well as the bright-dark, singular and other soliton solutions of the （2＋1）-dimensional nonlinear cubic-quintic Ginzburg-Landau equation. Comparing the solutions of this nonlinear equation together with each other are presented. Comparing our new results obtained in this article with the well-known results are given too.

New solitary wave in shallow water,plasma and ion acoustic plasma via the GZK-BBM equation and the RLW equation

This work obtains the disguise version of exact solitary wave solutions of the generalized(2+1)-dimensk>nal Zakharov-Kuznetsov-Benjamin-Bona-Mahony and the regu­larized long wave equation with some free parameters via modified simple equation method(MSE).Usually the method does not give any solution if the balance number is more than one,but we apply MSE method successfully in different way to carry out the solutions of nonlinear evolution equation with balance number two.Finally some graphical results of the velocity profiles are presented for different values of the material constants.It is shown that this method,without help of any symbolic computation,provide a straightforward and powerful mathematical tool for solving nonlinear evolution equation.

Dispersive Solitary Wave Solutions of Strain Wave Dynamical Model and Its Stability

In the materials of micro-structured, the propagation of wave modeling should take into account the scale of various microstructures. The different kinds solitary wave solutions of strain wave dynamical model are derived via utilizing exp(-φ(ξ))-expansion and extended simple equation methods. This dynamical equation plays a key role in engineering and mathematical physics. Solutions obtained in this work include periodic solitary waves, Kink and antiKink solitary waves, bell-shaped solutions, solitons, and rational solutions. These exact solutions help researchers for knowing the physical phenomena of this wave equation. The stability of this dynamical model is examined via standard linear stability analysis, which authenticate that the model is stable and their solutions are exact. Graphs are depicted for knowing the movements of some solutions. The results show that the current methods, by the assist of symbolic calculation, give an effectual and direct mathematical tools for resolving the nonlinear problems in applied sciences.

Novel solitary wave solution in shallow water and ion acoustic plasma waves in-terms of two nonlinear models via MSE method

By using modified simple equation method,we study the generalized RLW equation and symmetric RLW equation,the subsistence of solitary wave,periodic cusp wave,periodic bell wave solutions are obtained.We establish some conditions on the parameters for which the obtained solutions are dark or bright soliton.The proficiency of the methods for constructing exact solutions has been established.Finally,the variety of structure and graphical representation makes the dynamics of the equations visible and provides the mathematical foundation in shallow water,plasma and ion acoustic plasma waves.

Numerical simulation of parachute Fluid-Structure Interaction in terminal descent

A numerical simulation method for parachute Fluid-Structure Interaction （FSI） problem using Semi-Implicit Method for Pres- sure-Linked Equations （SIMPLE） algorithm is proposed. This method could be used in both coupling computation of para- chute FSI and flow field analysis. Both fiat circular parachute and conical parachute are modeled and simulated by this new method. Flow field characteristics at various angles of attack are further simulated for the conical parachute model. Compari- son with the space-time FSI technique shows that this method also provides similar and reasonable results.