Question

Use Exercise 34 to solve the initial-boundaryvalue problem. In some of these exercises Theorem 11.3.5(d) or Exercise 11.3.50(b) will simplify the computation of the coefficients in the mixed Fourier sine series. $$ \begin{array}{l} u_{t t}=9 u_{x x}, \quad 00 ,\\\ u(0, t)=0, \quad u_{x}(1, t)=0, \quad t>0 ,\\\ u(x, 0)=x^{2}(3-2 x), \quad u_{t}(x, 0)=0, \quad 0 \leq x \leq 1. \end{array} $$

Short answer

Answer: No, a nontrivial solution does not exist for the given wave equation using the separation of variables method with the given boundary conditions.

Step-by-step solution

Separation of Variables

Assume that the solution, \(u(x, t) = X(x)T(t)\). Substitute \(u(x, t)\) into the given equation and divide both sides by \(X(x)T(t)\) to get: $$\frac{T''(t)}{9T(t)}=\frac{X''(x)}{X(x)} = -\lambda$$ where \(-\lambda\) is the separation constant. Now, we have two ordinary differential equations for \(X(x)\) and \(T(t)\): $$X''(x) + \lambda X(x) = 0$$ and $$T''(t) = -9\lambda T(t)$$

Apply Boundary Conditions

First, consider the boundary condition \(u(0, t) = X(0)T(t) = 0\), which holds only when \(X(0) = 0\). Also, consider the boundary condition \(u_x(1, t) = X'(1)T(t) = 0\), which holds only when \(X'(1) = 0\). For \(X''(x) + \lambda X(x) = 0\), it is essential to distinguish the cases where \(\lambda\) is positive, zero, or negative. 1. When \(\lambda > 0\): \(X(x) = A\cos(\sqrt{\lambda} x) + B\sin(\sqrt{\lambda} x)\). Applying boundary conditions, we get \(A = 0\) and \(B\sqrt{\lambda}\cos(\sqrt{\lambda}) = 0\). For nontrivial solutions, there is no value for \(\sqrt{\lambda}\) that would satisfy the equation simultaneously. 2. When \(\lambda = 0\): \(X(x) = Ax + B\). Applying boundary conditions, we get \(A = B = 0\). It will not give nontrivial solutions as well. 3. When \(\lambda < 0\): \(X(x) = A\cosh(\sqrt{-\lambda} x) + B\sinh(\sqrt{-\lambda} x)\). Applying boundary conditions, it is impossible for this case to satisfy both boundary conditions and give any nontrivial solutions. Thus, the problem has no nontrivial solutions.

Solve for the Coefficients

As no nontrivial solutions can be obtained through this method, the given wave equation cannot be solved using separation of variables with these boundary conditions.

Construct the Final Solution

Since we were unable to find a series solution using the separation of variables approach for this wave equation with the given boundary conditions, no finite solution exists for this problem.

Key concepts

Fourier Sine Series

The Fourier Sine Series is a mathematical tool used to represent a function defined on a specific interval as an infinite sum of sine functions. This technique is particularly useful for solving problems involving partial differential equations, especially when the boundary conditions imply that the function equals zero at the endpoints.

In the context of the separation of variables, the Fourier Sine Series helps in expanding a solution in terms of sine functions that naturally meet the boundary conditions. For example, if you have a fixed endpoint, you can use a sine series to simplify and solve the problem. These series are often used in problems involving heat and wave equations, where the conditions at the boundaries play a crucial role.

By decomposing a function into its sine components, we can transform a complex problem into an easier one by focusing on individual modes determined by the eigenvalues.

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are equations containing a function and its derivatives. They are called "ordinary" to distinguish them from partial differential equations, which involve multiple variables.

In solving ODEs using separation of variables, we assume the solution takes a specific form, like a product of functions, each dependent on a single variable. This assumption allows us to separate the ODE into simpler equations.

• For example, if we have a differential equation like \( u_{tt} = 9u_{xx} \), separating variables lets us break it down into \( T''(t) = -9\lambda T(t) \) and \( X''(x) + \lambda X(x) = 0 \).

These simpler equations can often be solved more easily, providing insights into the behavior of the original equation.

The solutions to these ODEs often involve constants, determined by initial and boundary conditions, explaining the function's behavior over time or space.

Boundary Conditions

Boundary conditions are essential constraints that provide additional information needed to solve a differential equation. They specify the values that a solution must satisfy at the boundaries of the domain.

In the context of separation of variables, boundary conditions ensure unique solutions and often drive the form of the function expansions, such as Fourier sine or cosine series.

• For example, if \( u(0, t) = 0 \) and \( u_x(1, t) = 0 \), these conditions impose restrictions on the form of \( X(x) \), the spatial part of the solution.

Applying these conditions involves checking the behavior of the solution at the boundaries and ensuring any chosen constants meet these requirements.

The inability to satisfy boundary conditions can lead to no nontrivial solutions, as seen when boundary and initial conditions do not align well with the structure of a differential equation. Proper handling and understanding of these conditions are crucial for successfully solving and interpreting solutions to differential equations.