###### Conservation laws for a Boussinesq equation.

as those found by symmetry methods) can play an important role in the design and testing of numerical integrators; these solutions provide an important practical check on the accuracy and reliability of such integrators, [ 8 ]. It is known that conservation laws play a significant role in the solution process of an equation or a system of differential equations. Although not all of the conservation laws of partial differential equations (PDEs) may have physical interpretation they are essential in studying the integrability of the PDEs. For variational problems

###### Solutions and conservation laws of a generalized second extended (3+1)-dimensional Jimbo-Miwa equation

x y + k u y u x x + h ( u x t + u y t + u z t ) − k u x z = 0 , $$\begin{array}{} \displaystyle u_{xxxy}+ k \left( u_{y} u_{x}\right) _x + h (u_{xt} +u_{yt}+u_{zt})-k u_{xz} = 0 , \end{array} $$ (3) where h and k are constants. We obtain exact solutions of (3) using symmetry reductions along with simplest equation method. Furthermore, we derive conservation laws for (3) using the conservation theorem due to Ibragimov. Lie symmetry theory, originally developed by Marius Sophus Lie (1842-1899), a Norwegian mathematician, around the middle of the

###### On optimal system, exact solutions and conservation laws of the modified equal-width equation

method was applied to (1) and numerical solution of the MEW equation was obtained in [ 3 ]. In our study we use an entirely different approach to obtain new exact travelling wave solutions, namely cnoidal and snoidal wave solutions of MEW Equation (1) . Moreover, for the first time we derive conservation laws of the MEW equation by employing both the Noether approach as well as the multiplier approach. 2 Exact solutions of (1) constructed on optimal system In this section, we first compute Lie point symmetries of (1) and then use them to construct an optimal

###### Travelling waves and conservation laws of a (2+1)-dimensional coupling system with Korteweg-de Vries equation

scientific fields and in the theory of integrable systems. It describes the unidirectional propagation of long waves of small amplitude and has a lot of applications in a number of physical contexts such as hydromagnetic waves, stratified internal waves, ion-acoustic waves, plasma physics and lattice dynamics [ 2 ]. Equation (1) has multiple-soliton solutions and an infinite number of conservation laws and many other physical properties. See for example [ 3 , 4 , 5 ] and references therein. Recently focus has shifted to the study of coupled systems of Korteweg-de Vries

###### Multiplier method and exact solutions for a density dependent reaction-diffusion equation

leading to the integration by quadrature of ordinary differential equations, to the determination of invariant solutions of initial and boundary value problems and to the derivation of conservation laws [ 2 ]. In some particular cases this equation has been studied by other authors. In the particular case of g ( u ) = 1 [ 9 ], symmetry reductions and exact solutions were obtained using classical and nonclassical symmetries. u t = u ( 1 − u ) + 1 x x u u x x , $$\begin{array}{} \displaystyle u_t=u(1-u)+\frac{1}{x}\left[xuu_x\right]_x, \end{array}$$ (2) by

###### Musculoskeletal Outcomes from Chronic High-Speed High-Impact Resistive Exercise

and conservation. Journal of Musculoskeletal and Neuronal Interactions 15, 215-226. Mittag U Kriechbaumer A Bartsch M Rittweger J 2015 Forms follows function: A computational simulation exercise on bone shape forming and conservation Journal of Musculoskeletal and Neuronal Interactions 15 215 226 Nguyen VH, Loethen J, LaFontaine T (2008) Resistance training and dietary supplementation for persons with reduced bone mineral density. Strength and Conditioning Journal 30, 28-31. 10.1519/SSC.0b013e318174d6f8 Nguyen VH Loethen J

###### New Complex and Hyperbolic Forms for Ablowitz–Kaup–Newell–Segur Wave Equation with Fourth Order

Nonlinear Sciences 2 2 403 414 2017 [76] C. Ünlükal, M. Şenel, B. Şenel (2018). Risk Assessment with Failure Mode and Effect Analysis and Gray Relational Analysis Method in Plastic Enjection Prosess, ITM Web of Conferences, 22(01023), 1-10, 2018. Ünlükal C. Şenel M. Şenel B. 2018 Risk Assessment with Failure Mode and Effect Analysis and Gray Relational Analysis Method in Plastic Enjection Prosess ITM Web of Conferences 22 01023 1 10 2018 [77] C.M.Khalique, I.E.Mhlanga, Travelling waves and conservation laws of a(2+1)-dimensional coupling system with Korteweg-de Vries

###### New Complex Hyperbolic Structures to the Lonngren-Wave Equation by Using Sine-Gordon Expansion Method

, Ukraine 24 26 November 2016 [44] C.M.Khalique, I.E.Mhlanga, Travelling waves and conservation laws of a (2 + 1)-dimensional coupling system with Korteweg-de Vries equation, Appl. Math. Nonlinear Sciences 3(1) (2018) 241-254 10.21042/AMNS.2018.1.00018 Khalique C.M. Mhlanga I.E. Travelling waves and conservation laws of a (2 + 1)-dimensional coupling system with Korteweg-de Vries equation Appl. Math. Nonlinear Sciences 3 1 2018 241 254 [45] Baskonus, H.M.; Koc, D.A.; Bulut, H. New travelling wave prototypes to the nonlinear Zakharov-Kuznetsov equation with power law

###### Fully discrete convergence analysis of non-linear hyperbolic equations based on finite element analysis

of resistance and inductance per unit transmission line can be used to describe the characteristics of dielectrics. According to the law, a set of telegraph equations can be established, which can be simplified to a standard wave equation without loss [ 6 ]. In addition, hydrodynamic problems in aviation, meteorology, ocean, petroleum exploration and other fields are reduced to solving non-linear hyperbolic partial differential equations (PDEs; known as conservation laws in foreign literature). The basic difficulty of this kind of equation is that the solution

###### Symmetry Reductions for a Generalized Fifth Order KdV Equation

nonlinear systems. Furthermore, once the Lie symmetries and their corresponding group invariant solutions have been obtained we have transformed the equation ( 3 ) into an ordinary differential equation and we have applied the double reduction method given by Sjöberg [ 23 ]. That way, we have solved the ordinary differential equation (ODE), whose solutions provide solutions of the original partial differential equation ( 3 ). Examples about the method are in [ 12 , 6 ] The double reduction method emphasizes and uses the relation between symmetries and conservation