Explore chapters and articles related to this topic
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.
Numerical Solution of Boundary-Value Problems
Published in Ramin S. Esfandiari, Numerical Methods for Engineers and Scientists Using MATLAB®, 2017
The finite-difference method is the most commonly used alternative to the shooting method. The interval [a, b] over which the BVP is to be solved is first divided into N subintervals of length h = (b − a)/N. As a result, a total of N + 1 grid points are generated, including x1 = a and xN+1 = b, which are the left and right endpoints. The other N − 1 points x2, …, xN are the interior grid points. At each interior grid point, the derivatives involved in the differential equation are replaced with finite divided differences. This way, the differential equation is transformed into a system of N − 1 algebraic equations that can then be solved using the methods previously discussed in Chapter 4.
Boundary-Value Problems and Integral Equations
Published in Azmy S. Ackleh, Edward James Allen, Ralph Baker Kearfott, Padmanabhan Seshaiyer, Classical and Modern Numerical Analysis, 2009
Azmy S. Ackleh, Edward James Allen, Ralph Baker Kearfott, Padmanabhan Seshaiyer
In the shooting method, we solve a boundary value problem by iteratively solving associated initial value problems. Consider () (a)y″(x)+p(x)y′(x)+q(x)y(x)=f(x),(b)y(0)=α,y(1)=β,
Melting heat transfer of MHD stagnation point flow of non-Newtonian fluid due to an elastic sheet
Published in International Journal of Ambient Energy, 2022
Mahantesh M. Nandeppanavar, M. C. Kemparaju, N. Raveendra
In the present study Casson fluid flow and Casson rheological model accurately predicts mud rheology and offers many advantages over the yield power law, the Bingham plastic and the power law rheological models because it more accurately characterises mud behaviour across the entire shear rate conditions and also the Buongiorno model is later adopted to capture the Brownian motion and thermophoresis influences with the presence of nanofluids. The Buongiorno model clearly explains the combined effects of thermophoresis and Brownian on the physical motion and also the action of combined Brownian motion and thermophoresis is found to cool the system by enhancing the Nusselt number. We have to investigate the effect of magnetic field on the stagnation-point flow of Casson fluid. The shooting method is adapted to solve the coupled boundary value problem and this method gives more accurate results of the considered problem.
Analysis of auto cubic catalysis and nanoscale heat transport using the inclined magnetized Cross fluid past over the wedge
Published in Waves in Random and Complex Media, 2023
Assad Ayub, Zulqurnain Sabir, Salem Ben Said, Haci Mehmet Baskonus, Rafaél Artidoro Sandoval Núñez, R. Sadat, Mohamed R. Ali
There are several techniques [45–54] for solution of such problem. In this problem, the numerical procedures, the shooting method is applied to handle the boundary conditions and converted into initial value problems. Hence, bvp4c technique can be used to process further to solve the system of ODEs. Analytical and Matlab procedures [55–59] are given as The bvp4c is R–K 45 numerical method, which is used in the process of convergence to solve first-order linear/nonlinear ODEs, and this method is also known as collection procedure of order four. This numerical strategy has a key point to choose the finite and approximate values of η∞; in this case, η∞ = 5.
Peristaltic transportation of Carreau–Yasuda magneto nanofluid embedded in a porous medium with heat and mass transfer
Published in Waves in Random and Complex Media, 2022
Y. Akbar, J. Iqbal, M. Hussain, H. Khan, H. Alotaibi
The Mathematica software is employed to compute numerical solutions of (22), (23), and (25) with boundary conditions (28) via built-in command NDSolve. NDSolve is built in shooting method. This approach ensures the accuracy of finding a solution to the boundary value problem using an appropriate step size. In the sense of the Shooting method, the boundary conditions for the resulting system are transformed into the initial conditions. Therefore, the prescribed equations to be tackled is as follows: