Solving Heat Flow Problem: Initial Boundary Value Problem

In summary, the conversation is about solving an initial boundary value problem with the equation du/dt=5(d^2u/dx^2) and the given conditions. The proposed solution involves using a Fourier series equation with the coefficients a0 and an, and possibly adjusting the approach by using the sine series instead of the cosine series to satisfy the boundary condition. The next steps involve plugging values into the general u(x,t) equation.
  • #1
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Homework Statement



Find a formal solution to the given initial boundary value problem.

du/dt=5(d^2u/dx^2) 0<x<1 t>0
u(0,t)=u(1,t)=0 t>0
u(x,0)=(1-x)(x^2) 0<x<1

Homework Equations



1) u(x,t) = a0/2 + sum[an*e^(-b(n pi/L)^2*t) * cos(n pi x/L)

2) Fourier series equation

The Attempt at a Solution



(1-x)(x^2) = a0/2 + sum(an * cos(n pi x) with cn = an

I calculate a0=1/6

an = 2* integral[(1-x)(x^2)(cos n pi x)dx] from 0 to 1


I'm wondering if this is write so far? And if so, how do I proceed from here? Do I just plug everything back into the general u(x,t) equation?

Thanks!
 
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  • #2
You might not want to use the cosine series. Since one of your BC's is u(0,t) = 0. You will never satisfy that condition when cos(0) = 1. Try using the sine series instead.
 

What is a heat flow problem?

A heat flow problem is a mathematical model used to describe the transfer of thermal energy from one point to another. It involves solving a set of differential equations to determine the temperature distribution within a system over time.

What is an initial boundary value problem?

An initial boundary value problem is a type of problem that involves finding a solution to a differential equation that satisfies both initial conditions, which describe the system at a particular starting point, and boundary conditions, which describe the behavior of the system at its boundaries.

What are the main steps in solving a heat flow problem?

The main steps in solving a heat flow problem include: defining the problem, setting up the initial and boundary conditions, choosing an appropriate mathematical model, solving the differential equations using numerical or analytical methods, and analyzing and interpreting the results.

What are some common techniques used to solve heat flow problems?

Some common techniques used to solve heat flow problems include the finite difference method, the finite element method, and the method of separation of variables. These methods involve breaking down the problem into smaller, more manageable parts and then using numerical or analytical techniques to solve them.

Why are heat flow problems important in science and engineering?

Heat flow problems are important in science and engineering because they allow us to understand and predict how thermal energy is transferred in various systems. This information can be used to design and optimize processes and systems, such as in the fields of heat transfer, thermodynamics, and materials science.

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