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Numerical Analysis

Simply supported beam: Maximum bending moment

The bending moment M at position x m from the end of a simply supported beam of length L m carrying a uniformly distributed load of w Kn m-1 is given by M = w/2 L (L-x) - w/2 (L - x)^2 Show that the maximum bending moment occurs at the mid point of the beam, and determine its value in terms of w and L.

Heat Equation Problem

Find the solution u(x,t) of the heat equation: ut = 1/2 uxx (a) with initial data u(x,0) = x (b) with initial data u(x,0) = x^2 (c) with initial data u(x,0) = sinx (d) with initial data u(x,0) = 0 x < 0 and u(x,0) = 1 or x >/= 0 I know the solution of the heat equation with given initial data is unique. So if you happe

The 1D heat equation

Please see the attached file for the fully formatted problems. Solve the heat equation u_t = ku_xx for 0<x<L With boundary conditions u(0,t)=u(L,t)=0 Solve for the initial value conditions: a. u(x,0) = sin(5*Pi*x/L) b. u(x,0) = x c. For part b, plot the solution at t=0, 0.1, 1

Gradient Fields

Please see the attached file for the fully formatted problems. find a function f(x,y) such that the Grad(f) = (2xy+y^3+1, x^2+3xy^2) Explain why you cannot find f(x,y) such that Grad(f) = ( x^2+3xy^2, 2xy+y^3+1)

Sturm-Liouville: Expansion in Eigenfunctions

Hello, I need a detailed solution of the attached problem. Consider the eigenvalue problem y" + lambda^2 * y = 0 With the boundary conditions: y(0) = y'(1)=0 1. Determine the eigenfunctions and eigenvalues. 2. show that the eigenfunctions are orthogonal. 3. Show how to expand a smooth function in terms of the

Examples of the Method of Characteristics for Solving PDEs

1. Use the method of characteristics to solve the advection equation du/dt=-kdu/dx-ru subject to the initial condition u(x,0)=f(x). 2. Use the method of characteristics to solve du/dt+te^(-t^2))du/dx=usin(t) subject to the initial condition u(x,0)=e^(-x^2)) (See attachment for the above questions formatt

Cancellation Round-off Error : Numerical Analysis : Confirmation

I have numerically solved the following quadratic equation: 1.002x2 - 11.01x + 0.01265 = 0. IS IT POSSIBLE THAT IN THIS INSTANCE EQUATION (2) EQUALS (2A) BELOW: If b2 - 4ac >0, the quadratic equation ax2 + bx +c = zero has two real solutions x1, x2 given by the typical: (1) x1 = (-b + sqrt(b^2-4ac))/ (2a)

Solve: Boundary Value Problem

Boundary Value Problem. See attached file for full problem description. Consider the boundary value problem: PDE: u_t = u_xx, (0 < x < 4) BCs: u_x(0, t) = -2, u_x(4, t) = -2 ICs: u(x, 0) = {0 if 0 <= x <= 2 {2x - 4 if 2 <= x <= 4 (a) Find the steady-state solution. (NOTE: There are Neumann BCs at e

PDE by SL method

See attached file for full problem description. Solve by Eigenfunction.

Actual, Absolute and Maximum Relative Errors in Data

A touring company has the following box office takings for the recent month. Venue Clarion West Riding West End Burlington Huntingdon Erie Adelphia Wilson Reading How do I estimate the abosolute and relative maximum and average errors in the total of the above rounded data? How do I determine the actual error i

Mixture Problem as a Differential Equation.

Please solve using separation of variables method. A certain chemical is converted into another chemical by a chemical reaction. The rate at which the first chemical is converted is proportional to the amount of this chemical present at any instant. Ten percent of the original amount of the first chemical has been convert

Breaking an Eigenvalue Equation into Domains

Examine the eigenvalue equation below and then break the domain into four different regions (like a>0) and b=0 is one such domain. Describe the behavior of the equation in each domain.... Please see the attached file for the fully formatted problems. keywords: differential equations, trace, node

Adam collects stamps.

Adam collects stamps. He has 18 bird stamps,9 flower stamps and 12 butterfly stamps. For a school project, he will display an equal number of each kind of stamp on a small poster boards. What is the greatest number of poster boards Adam can make if he uses all of the stamps?

Numerical Analysis Problem/Secan Method

(See attached file for full problem description with proper symbols) --- A) Consider a variation of Newton's method in which only one derivative is needed; that is . Find C and s such that . B) Find the conditions on to ensure that the iteration will converge linearly ( ) to a zero of f if started near the zero.

Numerical Stability

(See attached file for full problem description with equations) --- There is a function f of the form for which and . Determine and , and assess the sensitivity of these parameters to slight changes in the values of f at the two indicated points. ---

Solving an IVP with Maple: Euler and Improved Euler Method

Use Maple to solve this exercise: Consider the following (IVP) logistic model p' = 10p(1-p) with p(0)=0.1 1. Solve this IVP and graph the solution over the interval [0, 10], Write down the Euler approximation, and Improved Euler approximation with step size h. 2. Compute and plot the first 100 points of the Euler method

Solving 2 ODE's with One Unknown

Given that: dMN/dt = P¬oert - pMN + m(MS - MN) dMS/dt = m(MN - MS) - pMS¬ MN¬ (0) = MS (0) = 0 And using: M¬N + MS = [Po/(r + p)](ert - e-pt) Show that MS(t) = And show that R = MN(t) / MS(t) = Please see the attached file for the fully formatted problems.

PDE : Riemann's Method for Solving Cauchy Problem

Hello. Thanks for help! I will use * to indicate a partial derivative. For example, u*x denotes the partial derivative of u with respect to x. This is the probelm: Use Riemann's method to solve the Cauchy problem: u*xx + 4u*xy +3u*yy = 1, u=1 and u*n = square root of 5 times x, on the intial curve y=2x. If this

Numerical Methods, 400 Undergraduate level.

The problem is from Numerical Methods. Please show each step of your solution and tell me the theorems, definitions, etc. if you use any. If there is anything unclear in the question, let me know. Thank you. (Complete problem in attachment)

Numerical Methods: Fixed-Point Iteration

Please show each step of your solution and tell me the theorems, definitions, etc. if you use any. Let g(x) = 0.5x + 1.5 and p0 = 4, and consider fixed-point iteration. a) Show that the fixed point is P=3 b) Show that |P - Pn| = |P - Pn-1|/2 for n = 1, 2, 3... c) Show that |P - Pn| = |P - P0|/2^2 for n = 1, 2, 3...

Three=digit rounding arithmetic

The problem is from Numerical Methods. Please show each step of your solution and tell me the theorems, definitions, etc. if you use any. 7. Use three-digit rounding arithmetic to compute the following sums... Please see attached.