Question

and hence one obtains (10.122). Let \(u_{1}(x, t), u_{2}(x, t), u_{3}(x, t)\), respectively, be solutions of the problems (for \(00)\) \(u_{t}-u_{x x}=F(x, t), \quad u(0, t)=0, \quad u(\pi, t)=0 ;\) \(u_{t}-u_{x x}=0, \quad u(0, t)=a(t), \quad u(\pi, t)=0 ;\) \(u_{t}-u_{x X}=0, \quad u(0, t)=0, \quad u(\pi, t)=b(t) .\) Show that \(u_{1}(x, t)+u_{2}(x, t)+u_{3}(x, t)\) is a solution of the problem $$ u_{t}-u_{x x}=F(x, t), \quad u(0, t)=a(t), \quad u(\pi, t)=b(t) . $$ This shows that the effects of the different ways of forcing the system combine by superposition.

Short answer

Question: Show that if \(u_1(x,t)\), \(u_2(x,t)\), and \(u_3(x,t)\) are solutions of different heat equation problems with the forcing function F(x,t), then the sum of these three solutions, \(u(x,t)=u_1(x,t)+u_2(x,t)+u_3(x,t)\), is also a solution of another heat equation problem with different boundary conditions. Answer: The sum \(u(x,t)=u_1(x,t)+u_2(x,t)+u_3(x,t)\) is a solution of the problem \(u_t - u_{xx} = F(x,t)\) with boundary conditions \(u(0,t) = a(t)\) and \(u(\pi,t) = b(t)\), where the effects of the different ways of forcing the system combine by superposition.

Step-by-step solution

Sum the given PDEs

Firstly, we'll sum up the left-hand sides of the given partial differential equations for \(u_1(x,t), u_2(x,t),\) and \(u_3(x,t)\): $$ (u_1)_t - (u_1)_{xx} = F(x,t)\\ (u_2)_t - (u_2)_{xx} = 0\\ (u_3)_t - (u_3)_{xx} = 0 $$ When summed together, we get: $$ [(u_1)_t + (u_2)_t + (u_3)_t] - [(u_1)_{xx} + (u_2)_{xx} + (u_3)_{xx}] = F(x,t) $$

Define the sum

Now, let's define \(u(x,t) = u_1(x,t) + u_2(x,t) + u_3(x,t)\). Then, we can rewrite the equation from Step 1 as: $$ u_t - u_{xx} = F(x,t) $$

Check boundary conditions

Now, we'll check the boundary conditions of the given problem: $$ u(0,t) = u_1(0,t) + u_2(0,t) + u_3(0,t) = 0 + a(t) + 0 = a(t) $$ and $$ u(\pi,t) = u_1(\pi,t) + u_2(\pi,t) + u_3(\pi,t) = 0 + 0 + b(t) = b(t) $$

Conclude

Since the sum \(u(x,t)\) satisfies both the partial differential equation and the boundary conditions of the given problem, we can conclude that \(u(x,t)=u_1(x,t)+u_2(x,t)+u_3(x,t)\) is a solution of the problem: $$ u_t - u_{xx} = F(x,t), \quad u(0,t) = a(t), \quad u(\pi,t) = b(t) $$ This proves that the effects of the different ways of forcing the system combine by superposition.

Key concepts

Partial Differential Equations

Partial Differential Equations (PDEs) are mathematical equations that involve multiple independent variables, and an unknown function that depends on these variables. A PDE specifies relationships between partial derivatives of the unknown function. In the context of the exercise, you encounter a PDE that models phenomena varying with both space and time: \[ u_{t} - u_{xx} = F(x, t). \]This equation describes how the function \( u(x, t) \) evolves over time \( t \) and space \( x \). The term \( u_{xx} \) represents the second derivative with respect to space, indicating diffusion or spreading, while \( u_t \) indicates how \( u \) changes over time. Understanding PDEs is crucial because they describe many physical phenomena, like heat and wave propagation.

Boundary Conditions

Boundary conditions are constraints necessary to determine a unique solution for PDEs. They specify the behavior of the solution at the borders of the domain. In this problem, the domain is the interval \( [0, \pi] \).

• For \( u_1(x, t) \), the boundary conditions are \( u(0, t) = 0 \) and \( u(\pi, t) = 0 \). This means the function is fixed at zero at both ends.


• For \( u_2(x, t) \), the boundary conditions are \( u(0, t) = a(t) \) and \( u(\pi, t) = 0 \). The function has a varying value \( a(t) \) at \( x = 0 \), while fixed at zero at \( x = \pi \).


• For \( u_3(x, t) \), \( u(0, t) = 0 \) and \( u(\pi, t) = b(t) \), meaning the function is zero at \( x = 0 \) and takes on the value \( b(t) \) at \( x = \pi \).

This setup ensures the uniqueness of each function \( u_1, u_2, u_3 \), allowing us to build the complete solution \( u(x, t) \) using their superposition.

Solution Collation

Solution collation refers to combining individual solutions to form a complete solution to a given PDE. This is achieved by superposing the solutions: \[ u(x, t) = u_1(x, t) + u_2(x, t) + u_3(x, t). \]

• The solution \( u_1(x, t) \) satisfies the PDE with specified boundary conditions.
• Both \( u_2(x, t) \) and \( u_3(x, t) \) address the homogenous PDE \( u_t - u_{xx} = 0 \), but with their unique boundary handles \( a(t) \) and \( b(t) \).

The combination \( u(x,t) \) effectively meets the original problem's differential equation and boundary conditions. This method effectively represents the system's response due to each part of the boundary, demonstrating the superposition principle.

Mathematical Proof

A mathematical proof is a logical argument demonstrating the validity of a stated proposition. Here, the task was to prove that the sum \( u(x, t) = u_1(x, t) + u_2(x, t) + u_3(x, t) \) is a solution of the original problem.To begin the proof, observe that by summing the PDEs for \( u_1, u_2, \) and \( u_3 \), the terms involving zero simplify, leaving us with the original non-homogeneous equation: \[(u_1)_t - (u_1)_{xx} + (u_2)_t - (u_2)_{xx} + (u_3)_t - (u_3)_{xx} = F(x, t).\] Checking the boundary conditions confirms:- \( u(0, t) = a(t) \)- \( u(\pi, t) = b(t) \).These match the problem requirements, thereby successfully proving that \( u(x, t) = u_1(x, t) + u_2(x, t) + u_3(x, t) \) adheres to both the PDE and boundary conditions. Hence, the superposition principle holds, allowing us to conclude the proof.