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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)

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

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.

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

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...

Partial Differential for Diffusion Equation

Suppose that u(x,t) satisfies the diffusion equation... for 0<x<L and t>0, and the Robin boundary conditions... where k, L, a0 and aL are all positive constants. Show that... is a decreasing function of t. Please see attached for full question.

Heat Equation, Boundary & Steady State Conditions, Initial Value

Please help with the following problems involving numerical analysis. (a) Find all the separated solutions of the attached heat equation (satisfying the attached boundary condition) (b) Use these separated solutions to write a series solution for the initial value problem posed by the attached pde and the attached boundar

Initial Value Problem (IVP); Euler's Method; Step Sizes

4. Consider the initial value problem (IVP): y'(t) = 3+t+y y(0)=1 a) Approximate y(1) using Euler's method and step sizes of 0.2. Perform these calculations by hand. What is the exact value of y(1)? b) Use the computer (e.g. ODE Architect, ODE Toolkit, or your own program) to approximate y(1) using step sizes of 0.1, 0.05,

Numerical Euler's Method for Initial Value Problem

Consider the initial value problem (IVP): y'(t) = y^2 y(0)=1 Approximate y(1) using Euler's method and step sizes of 0.25. Perform these calculations by hand (using a calculator for arithmetic is ok). What is the true value of y(1)?

Carrying capacity of a biologists

Biologists stocked a lake with 400 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 7000 . The number of fish doubled in the first year. dP/dt = rP(1- p/K) find an expression for the size of the population after t years by determining constant r.

Bisection and newton's method

Q Using the bisection method, find the positive root of 2x(1 + x^2)^-1 = arctan x. Using this root as x0; apply Newton's method to the function f(x) = arctan x: Interpret the results you obtain.

Elliptic Boundary Value Problem

Uxx means second derivative with respect to x Uyy means second derivative with respect to y Uxx + Uyy = 0, 0 < x < pi, 0 < y < pi U(x,0) = 0, U(x,pi) = 1, 0 < x < pi U(0,y) = 0, U(pi,y) = 1 0 < y < pi I know the problem has to be broken into 2 separate problems using U = V + W with zero conditions on 3 sides fo

Solve an IVP ODE using the Method of Variation of Parameters

Please see the attached file for the fully formatted problems. Solve an IVP ODE using the method of variation of parameters Find the solution of the system X' using the method of variation of parameters 2 0 0 cos(t) X' = -1 0 -1 X + sin(t) 1 1 2 e^-t that satisfies the int