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Find the solution of the heat conduction problem $$ \begin{aligned} u_{x x} &=4 u_{t}, \quad 00 \ u(0, t) &=0, \quad u(2, t)=0, \quad t>0 \ u(x, 0) &=2 \sin (\pi x / 2)-\sin \pi x+4 \sin 2 \pi x, \quad 0 \leq x \leq 2 \end{aligned} $$

Short Answer

Expert verified
Question: Determine the temperature distribution u(x, t) for the given heat conduction problem described by the partial differential equation with boundary and initial conditions: PDE: uxx=4ut, 0<x<2, t>0 Boundary Conditions: u(0,t)=0, u(2,t)=0 Initial Condition: u(x,0)=2sin(πx/2)sin(πx)+4sin(2πx) Answer: The temperature distribution u(x, t) for the given heat conduction problem is: u(x,t)=2eπ28tsin(πx2)e4π28tsin(πx)+4e16π28tsin(2πx).

Step by step solution

01

We assume a solution of the form u(x,t)=X(x)T(t). Plugging this into the PDE, we get: X(x)T(t)=4X(x)T(t). Now, we'll divide both sides by X(x)T(t), which yields: X(x)X(x)=4T(t)T(t). Since the left side is a function of x only and the right side is a function of t only, they must be equal to a constant. Let's call it λ2: X(x)X(x)=4T(t)T(t)=λ2. #Step 2: Solve for X(x)#

We now have two ordinary differential equations (ODE): X(x)+λ2X(x)=0. This is a second-order linear ODE, and we know that its general solution is: X(x)=Acos(λx)+Bsin(λx), where A and B are constants. #Step 3: Apply Boundary Conditions for X(x)#
02

We are given two boundary conditions for the PDE: u(0,t)=0,u(2,t)=0. Plugging these into our assumed solution and solving for X(x) yields: X(0)=Acos(0)+Bsin(0)=0A=0, and X(2)=Bsin(2λ)=0. As we don't want the trivial solution of X(x)=0, we need sin(2λ)=0, which implies λ=nπ/2 for nZ. Our final solution for X(x) is then: Xn(x)=Bnsin(nπx/2), where nZ and Bn is a constant that depends on n. #Step 4: Solve T(t) ODE#

Now let's solve the ODE for T(t): 4T(t)+λ2T(t)=0. Integrating, we get: T(t)=λ24T(t). The solution to this ODE is an exponential: Tn(t)=Cnen2π28t, where Cn is a constant that depends on n. #Step 5: General Solution & Apply Initial Condition#
03

Now that we have solutions for both Xn(x) and Tn(t), we can form the general solution as an infinite sum: u(x,t)=n=BnCnen2π28tsin(nπx/2). Applying the initial condition u(x,0)=2sin(πx/2)sin(πx)+4sin(2πx), we find the coefficients BnCn using a Fourier sine series expansion. Comparing with our initial condition, we get: B1C1=2,B2C2=1,B4C4=4. with all other BnCn=0. #Step 6: Final Solution#

Plugging the values of the coefficients BnCn into the general solution, we finally obtain the solution to the heat conduction problem: u(x,t)=2eπ28tsin(πx2)e4π28tsin(πx)+4e16π28tsin(2πx).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Partial Differential Equations
Partial Differential Equations (PDEs) are mathematical equations that involve functions and their partial derivatives with respect to various variables. They are crucial in modeling and analyzing dynamic systems found in physics and engineering. A PDE can describe phenomena such as heat conduction, wave propagation, and fluid dynamics. In this exercise, we explore a specific example of a PDE known as the heat equation, which models the distribution of heat in a given domain over time. The heat equation is expressed as: uxx=4ut Here, uxx represents the second spatial derivative (concerning the variable x), and ut is the first time derivative. Solving PDEs like this involves finding functions that satisfy the equation under the given conditions, known as boundary and initial conditions. These conditions provide necessary information about the behavior of the function at specific points, helping to determine a unique solution.
Boundary Conditions
Boundary conditions are an essential component of solving PDEs, as they specify the behavior of a function at the boundaries of the domain. In the context of the heat equation, boundary conditions are necessary to ensure a well-defined and unique solution over time. In this problem, the boundary conditions are given by:
  • u(0,t)=0
  • u(2,t)=0
These conditions describe how the system behaves at the edges x=0 and x=2, indicating that the temperature remains zero at these points for all times t>0. By applying these conditions to the general solution X(x), we ensure that any potential solutions satisfy these boundary constraints, thus aiding in the selection of appropriate functions. Furthermore, boundary conditions help in determining constants in the general form of the solution, allowing for the extraction of a specific solution that fits the entire problem.
Fourier Series Expansion
The Fourier series expansion is a powerful mathematical tool used to express a function as an infinite sum of sines and cosines. This technique is particularly useful in solving PDEs, as it allows us to handle complex boundary and initial conditions by decomposing them into simpler, well-understood trigonometric forms. In this exercise, the initial condition is given by the function: u(x,0)=2sin(πx2)sin(πx)+4sin(2πx) To match this condition with our solution, we use a Fourier sine series expansion. This involves writing the solution as a sum of terms of the form BnCnen2π28tsin(nπx2). By comparing this series with the provided initial condition, we determine the coefficients BnCn for each relevant n. Thus, Fourier series expansion helps us solve the PDE by matching the infinite series solution with the initial data provided.
Separation of Variables
The separation of variables is a method used to transform a PDE into simpler ordinary differential equations (ODEs), which are easier to solve. This approach assumes that the solution can be expressed as a product of functions, each dependent on a single variable. In this problem, we assume that the solution u(x,t) can be written as X(x)T(t). Substituting this expression into the original PDE, we separate the variables to obtain: X(x)X(x)=4T(t)T(t)=λ2 This technique reduces the PDE into two ODEs: one for X(x) and another for T(t). Once separated, these ODEs can be individually solved. X(x) satisfies the equation X(x)+λ2X(x)=0, whose solutions involve trigonometric functions. Meanwhile, T(t) solves the equation 4T(t)+λ2T(t)=0, with solutions taking an exponential form. The separation of variables simplifies tackling complex PDEs by breaking them down into manageable parts.

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Most popular questions from this chapter

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