Also, new cnoidal wave solutions are derived via a new hypothesis in the form of the Weierrtrass elliptic function. By means of the Jacobian elliptic function ansatz, the cnoidal and soliary wave solutions are obtained.

Four different analytical methods (the Jacobian elliptic function, Weierrtrass elliptic function, the traditional tanh method and the sech-square) are devoted for solving this equation. In this work, some novel analytic traveling wave solutions including the cnoidal and solitary wave solutions of the planar Extended Kawahara equation are deduced. Our mathematic aneurysm model proposes a good analogue to the real aneurysm and proved that this model includes soliton that is a non-decreasing wave propagation. An analysis of this model provided that the presence of the daughter aneurysm and the thinning of the aneurysm wall are positively correlated with a sharp increase in blood pressure in the aneurysm dome. Then, the mathematical formula was solved in numerical simulations by changing the size of the aneurysm and the elasticity of the aneurysm wall. Our aneurysmal model is created as a two-story aneurysm model for this point, thus namely the five-element Windkessel. A dumbbell-shaped aneurysm is the most dangerous aneurysm to easily rupture.

We simulate an aneurysm by an electric circuit consisting of three different impedances, resistance, capacitance and inductance. Since an aneurysm is an end-sack formed on the blood vessel, it functions as an unusual blood path that has characteristic features such as a reservoir and bottle neck orifice. Intracranial aneurysms are well known vascular lesions, which cause subarachnoid hemorrhages. We propose a theoretical model for the intracranial aneurysm based on the Windkessel-type steady blood flow. The Windkessel model, which is known as a successful model for explaining the hemodynamic circulation, is a mathematical model with a direct correspondence with the electric circuit. Numerical results depending on the physical plasma parameters are presented. The obtained solutions could be applied for investigating the characteristics of the dissipative higher-order solitary and cnoidal waves in electronegative plasmas. Also, the obtained solutions help many researchers understand the mechanisms underlying a variety of nonlinear phenomena that can propagate in optical fiber, physics of plasmas, fluid mechanics, water tank, oceans, and seas. All the obtained solutions are able to investigate many types of the dissipative traveling wave solutions such as the dissipative solitary and cnoidal waves.

The comparison between the obtained solutions is examined. Moreover, the Crank–Nicolson implicit finite difference method is introduced for analyzing the evolution equation numerically. Using a suitable ansatz and with the help of the exact solutions of the undamped Gardner Kawahara equation, two general high-accurate approximate analytical solutions are derived. Two novel analytical solutions to the damped Gardner Kawahara equation and its related equations are reported. This investigation can help all researchers interested in studying the characteristics of many nonlinear phenomena in various fields of science such as physics of plasmas, physics of fluids, nonlinear optics, Veins and capillaries, Oceans and seas, etc. Moreover, the maximum global residual error to the analytical approximations is estimated. The validity of all exact solutions is checked. In the fourth part, we focus our efforts on obtaining some approximations to forced EKE. After that, a general formula for the approximate solution to the forced undamped mKE is obtained. Also, a particular soliton solution to the forced damped mKE is obtained using a direct ansatz. At the beginning, two different general formulas with different accuracy to the forced damped mKE are derived. In the third part, the family of forced damped modified KE (mKE) is analyzed comprehensively via several different techniques in order to obtain several approximations. In the same part, two different general formulas to the analytical approximations to the forced damped KE are obtained. In the second part, the ansatz method is devoted for deriving some exact SWs and CWs solutions to the forced KE. In this part, we use some different hypotheses to derive many exact traveling wave solutions (including solitary waves (SWs) and cnoidal waves (CWs)). This work is divided into four main parts: in the first part, the ansatz method is employed to obtain some novel exact solutions to the integrable Extended KE (EKE). In this investigation, a set of novel exact and approximate analytic solutions to the family of the forced damped Kawahara equation (KE) are derived in detail.

Nuclear time nsync