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The Generalized Fourier Series Solutions of the Euler-Bernoulli Beam Equation
Published in Wen L. Li, Weiming Sun, Fourier Methods in Science and Engineering, 2023
The system consisting of the differential equation (4.1) and the boundary conditions (4.3) can be used to describe a range of boundary value problems in science and engineering. Specifically, for r=1, the system reduces to a0d2ydx2+a1dydx+a2y=f(x),x∈(a,b)
Continuous-Wave Silica Fiber Lasers
Published in Johan Meyer, Justice Sompo, Suné von Solms, Fiber Lasers, 2022
Johan Meyer, Justice Sompo, Sune von Solms
For the sake of simplicity, Erbium-doped gain medium is considered as a two levels system and unidirectional propagation is considered. The set of equations describing the fiber laser is a two-point boundary value problem. We used numerical methods described in the previous chapter to solve it. Here we use an algorithm involving a Runge-Kutta 4th order scheme along with a shooting method (Morrison et al. 1962; Ha 2001; Flannery et al. 1992) to solve the system of equations. The only value known at the left-hand side of the laser is the pump power as the laser boundary conditions don’t explicitly provide a value for laser power at the boundary but a relation between them. Two fist guesses are provided for laser power at z = 0 (left hand) in addition to the known value of the pump power. Then we propagate the fields step by step in the cavity using a Runge-Kutta 4th order method. Once we reach z = L, (right hand), we apply the boundary condition and then propagate the fields backwards until we reach z = 0. The values of signal power at z = 0 is then compared with the initial guess. If the difference is found to be higher than a convergence value, we refine the guess using a linear secant method and propagate the field again until convergence is achieved. The shooting method transforms a boundary value problem into an initial value problem which is easier to solve.
Energy Principles and Variational Methods
Published in J. N. Reddy, Theories and Analyses of Beams and Axisymmetric Circular Plates, 2022
Essential boundary conditions are also known as Dirichlet or geometric boundary conditions, and natural boundary conditions are known as Neumann or force boundary conditions. In general, only one element of each of the duality pairs may be specified at any point of the boundary. When neither element of a duality pair is specified but a relation is known between the elements of the pair, the boundary conditions are known as the mixed boundary conditions. In a given problem, only one of the four combinations given in Eq. (2.5.38) can be specified. Problems in which all of the boundary conditions are of essential type are called Dirichlet boundary-value problems, and those in which all of the boundary conditions are of natural type are called Neumann boundary-value problems. Mixed boundary-value problems are those in which both essential and natural boundary conditions are specified.
A new dispersive wave with Love-type waves in a microstructure due to an impulsive point source
Published in Waves in Random and Complex Media, 2023
Abhishek K. Singh, Anusree Ray, Richa Kumari
The method of Green’s functions is an important technique for solving boundary value problems. Many researchers have employed Green’s function in their study keeping in view its inevitable role in solving the problems regarding the generation of waves due to impulsive forces. Covert [9] discussed the method for finding the displacement corresponding to SH-waves by adopting the method of Green’s function. The propagation of SH-wave in a stratified heterogeneous layer of finite depth due to point source was studied by Vlaar [10]. Chattopadhyay and Kar [11] derived the dispersion relation for Love-type waves propagating in an isotropic inhomogeneous medium under initial stress due to an impulsive point source. Afterward, Green’s function for an infinite two-dimensional anisotropic magneto-electroelastic medium containing an elliptical cavity was analyzed by Jinxi et al. [12]. The dispersion relation for the SH-waves propagating due to an impulsive point source in a magneto-elastic monoclinic layer lying over a heterogeneous monoclinic half-space was deduced by Chattopadhyay and Singh [13]. Lately, Hou and Zhang [14] introduced an accurate and efficient method for the piezoelectric coated functional devices based on the three-dimensional Green's functions under a tangential point force.
Features of Casson liquid in non-linear radiative magnetized cylindrical media: a numerical solution
Published in Waves in Random and Complex Media, 2022
Khalil Ur Rehman, Maryum Khalil, Nabeela Kousar, Wasfi Shatanawi
The analytic solution of a system of equations with corresponding boundary conditions (11) – (14) cannot be obtained because they are non-linear and coupled. So we use a numerical technique named the shooting method with the RK-4 technique. The shooting technique is a method for solving a boundary value problem by reducing it to an initial value problem in numerical analysis. It entails solving the initial value problem for various initial conditions until a solution is discovered that also meets the boundary conditions of the boundary value problem. Besides this, one can assess the solution techniques to examine the fluid flow narrating differential equations in Refs. [27–41]. To tackle the system of ODEs (11) and (12) with the boundary conditions (13) and (14) by using the shooting method, first, we have to convert these equations into the system of first-order differential equations. For this purpose, let us suppose. where, prime indicates the differentiation w.r.t .The coupled non-linear ODEs (11) and (12) are then converted into the following system of five first-order differential-equations and the boundary conditions are transformed as given by:
Numerical study of non-Fourier heat transfer in MHD Williamson liquid with hybrid nanoparticles
Published in Waves in Random and Complex Media, 2022
Maryam Haneef, Sayer Obaid Alharbi
More rise in thermal conductivity of the fluid because of dispersion of hybrid nanoparticles is possible than the rise in thermal conductivity because of the dispersion of single-type nanoparticles. Due to this significant rise in thermal conductivity, researchers have published several works on this topic. For instance, nano and hybrid nanoparticles were considered to enhance the thermal performance of ethylene glycol by Nawaz et al. [7]. Using hybrid nanofluid, Rana et al. [8] investigate the thermal improvement in an incompressible Carreau–Yasuda boundary layer flow. To solve boundary value problems numerically, they applied the finite element method (FEM). The heat transfer properties of mixtures of ethylene glycol and hybrid nanoparticles are investigated by Nawaz et al. [9]. The finite element approach is used to solve complex mathematical models. Heat transfer in the MHD flow of hybrid nanoparticles in the presence of viscous dissipation and porous medium is studied by Ahmad et al. [10]. In order to boost thermal conductivity, Nallusamy et al. [11] developed a mixture of hybrid nanoparticles in aluminum Oxide and copper and fluid under varying proportions. Ghalambaz et al. [12] explored the effects of hybrid nanoparticles on the melting heat transfer in the fluid contained in an enclosure.