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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

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

### Convolution theorem

Please help working on these problems Please show all steps section 7.7 # 2, See attached Use the convolution theorem to obtain a formula for the solution to the given initial value problem...

### Numerical Analysis Problem using Matlab

Evaluate the following with Simpson's scheme: 4 times the integral from 0 to 1 of 1/(1+x^2) and 8 times the integral of (sqrt(1 - x^2) - x) from 0 to 1/sqrt(2). Provide the codes used and all the results and work.

### Matrix relative to standard basis

Let D: P5 |-> P5, Where D = d /dx . Find the matrix of the D relative to the standard basis, { 1,x, x^2,x^3,x^4,x^5}.

### Initial-Value Problem Solved

Please see the attached file for the fully formatted problem. Solve the IVP: y'' + 4y' +13y = 13t2 -5t +24 +e^-2t(sin 3t)

### Second-order homogeneous Euler equation

Please show all steps to solution. Solve the second-order homogeneous equation 4x^2y''-4xy'+3y=0, by applying the transformation v=lnx, x>0 This is a second-order Euler equation.

### Nonhomogeneous equation

Please show all steps to solution. a) Find a particular solution to the nonhomogeneous equation y''''-y'''-y''-y'-2y=8x^5 Write out the general solution. b) Use the solution to problem A to solve the initial-value problem y''''-y'''-y''-y'-2y=8x^5 where y(0)=y'(0)=y'

### Numerical analysis proof

The problem I need solved is attached. Please provide as much detail as possible, so I can understand. Thanks Recall that a given vector norm |X| the operator norm of matrix A is given by ......

### Doolittle LU decomposition

Please see the attached problem: Please give the complete solution, include reasoning and calculations used to arrive at answer.

### Explain the following in your own words: (a + b)2 ¹ a2 + b2. Give a numerical example to illustrate this.

Explain the following in your own words: (a + b)2 ¹ a2 + b2. Give a numerical example to illustrate this.

### Numerical Analysis : Approximating Functions and Neville's Iterated Interpolation

Please see the attached file for the fully formatted problems.

### 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

### Numerical Analysis : Newton's Method, Sequence Convergence and Multiplicity

Please see the attached file for the fully formatted problems.

### You are using Newton's method with deflation to find the roots...

Please see the attached file for the fully formatted problems. Newton's methods with deflation to find the roots used are determined.

### Diffusion Equation

Show that S(x,y,t)=S(x,t)S(y,t) satisfies the diffusion equation. S_t = k(S_xx + S_yy)

### 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)

### Differenetial Equations and the Lanchester Combat Model : Solutions and Phase Portraits

Please see the attached file for the fully formatted problems.

### Numerical Concepts : Definitions

Please explain or define these concepts. 1. Natural number 2. Multiplication 3. Subtraction 4. Closure for addition 5. Commutativity 6. Associativity 7. Distributivity 8. Closure for multiplication 9. Contrast commutativity and associativity

### 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

### Numerical Methods : Pade Approximation

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### Numerical Analysis : Orthogonal Polynomials and Least Squares Approximations

Following questions need to be answered: 2, 4, 6, 7, 8, 11, & 12. (page 502) See attached file for full problem description.

### Numerical Analysis- Approximation Theory- Discrete Least Squares

Please provide solutions for following questions 1, 2, 5, 8 and 13. Problems start on p 490. See attached file for full problem description

### Numerical Analysis

See attached file for full problem description.

### 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