The Final Infinite Interval And Exponential Gegenbauer Function

"This course is intended to highlight mathematical principles, concepts, and techniques that are often used in scientific applications and illustrate how these techniques are employed in the context of specific problems in physics, chemistry, and biology. Topics include mathematical

by the Trapezoidal rule is Etrap = O(h2 ) and by Simpson’s rule is ESimp = O(h4 ). i) For each of these numerical integration rules, what conditions are required on the integrand f so these error estimates are valid? ii) Suppose that the error using h = 5 × 10−3 is E0 = 1.19 × 10−4 when using either the Trapezoidal rule or Simpson’s rule. For both rules, ˆ estimate the error if an interval width of h = 1 × 10−3 is used. iii) The Matlab command [z, w] = gauleg(N); calculates the N Gauss-Legendre nodes z and weights w for the interval [−1, 1]. Show how z and w can be used to numerically calculate

Let us suppose that Hamiltonian of the system has a form $\hat{H}\big(p,q,\lambda \big)$. Here $p,q$ are canonical coordinates and $\lambda$ is the parameter. For the solution of Schrodinger equation $\imath \frac{\partial \Psi}{\partial t}=\hat{H}\Psi$ we implement following ansatz: $\Psi =\sum_{n}a_{n}\big(t\big)\varphi_{n}\big(p,q,\lambda\big)\exp \big\{-\imath \int_{-\infty}^{t}E_{n}\big(\lambda\big)dt\big\}$, where $E_{n}\big(\lambda\big)$ are the instantaneous quasi-energies that adiabatically depend on the parameter $\lambda$. After standard derivations for time dependent coefficients $a_{n}\big(t\big)$ we obtain iterative solution $a_{n}^{(1)}\big(t\big)=-\int_{-\infty}^{t}d\tau\sum_{m\neq n}\frac{\big\langle\varphi_{n}\big|\frac{\partial H}{\partial \lambda}\big|\varphi_{m}\big\rangle\dot{\lambda}}{E_{m}-E_{n}}\times a_{m}\big(-\infty\big)\exp \big\{-\imath \int_{-\infty}^{\tau}\big(E_{m}-E_{n}\big)d\acute{\tau}\big\}$. Adiabatic approximation is valid when the following criteria holds $\frac{a_{n}^{(2)}}{a_{n}^{(1)}}\sim \frac{\partial H}{\partial t}\frac{1}{\big(E_{m}-E_{n}\big)^{2}}$. Here $a_{n}^{(2)}$ is the second order correction to $a_{n}\big(t\big)$.

The central differencing method is used to find an expression for d2u/dx2 in the form ui-1 +ui+1

2. On the problems, include ALL computations, properly labeled in the space provided. Partial credit may be given at the discretion of the instructor.

The author did struggle a bit with all of the different functions and how to find the solutions to the problems

The following problem appeared in the February-March supplementary grade 12 examination of 2013 (paper 2). Please take note that this question is subsection of a question which already introduced some information. The original question number in the paper is 4.3.

Please complete the following 7 exercises below in either Excel or a word document (but must be single document). You must show your work where appropriate (leaving the calculations within Excel cells is acceptable). Save the document, and submit it in the appropriate week using the Assignment Submission button.

Results for 81×81grid with Re=1000 are shown in Figure 1.It contains contours and graphs at Re =1000, with convergence criteria =1e-6, No of iterations = 15000, and No of time steps =50000.

In this coursework I am going to investigate numerical methods of solving equations. The methods I will use are:

The initial activities were based around the identification of complex problems in general and some of the various models that can be utilised to provide

Based on the above analysis we suggest that the solution provided by Coopers and Myers are

In this section, we discuss all the previous work that has been done in this same field together with different hypothesis introduced. A full discussion of

The presence of a convective term means that the data of the problem across the entire domain has an influence on the outflow boundary layer, in contrast to reaction diffusion problems where it is only the data local to a boundary that has an influence on the boundary layer. Hegarty and O'Riordan \cite{115} constructed a parameter-uniform numerical methods for the singularly perturbed problem posed on a circular domain. Asymptotic expansions for the solutions to such a problem have been established in \cite{120,121,122,123}. Analytical expressions for the exact solution, in the case of constant data, are given in \cite{123} as a Fourier series with coefficients written in terms of modified Bessel functions. In \cite{120,121,122} sufficient compatibility conditions are identified so that the accuracy of the asymptotic expansion can be estimated in the $L^2-$norm or in a suitably weighted energy norm. These expansions are derived without recourse to a maximum principle. The smooth case, where significant compatibility is assumed, is examined in \cite{120}; the non-compatible case with a polynomial source term is studied in \cite{125} and for a more general source term in \cite{122}. Based on these asymptotic expansions, a numerical method is constructed in \cite{124} for the problem, which uses a quasi-uniform mesh (which is, hence, not layer-adapted). By enriching the finite element subspace, with certain exponential boundary-layer basis

There are many papers in the literature dealing with the numerical approximation of the solution for time dependent singularly perturbed reaction diffusion problems. In \cite{186,188,189}, 1D problems are considered, in \cite{185,187} the case of 2D parabolic problems is analyzed and coupled linear systems are solved in \cite{190}. Different techniques have been used to prove the uniform convergence of the numerical method defined for each problem. In some papers the totally discrete method is directly considered and the analysis of the convergence is based on appropriate bounds for the truncation error and the discrete maximum principle for the totally discrete operator. Nevertheless, this technique