Question

Solve the nonhomogeneous initial-boundary value problem. $$ \begin{array}{l} u_{t}=3 u_{x x}-18 x, \quad 00, \\ u_{x}(0, t)=-1, \quad u(1, t)=-1, \quad t>0, \\ u(x, 0)=x^{3}-2 x, \quad 0 \leq x \leq 1 \end{array} $$

Short answer

The particular solution, w(x, t), for the given problem is: $$ w(x, t) = 6x^3 - x $$

Step-by-step solution

Separate the Function u into Homogeneous and Particular Solutions

Let's rewrite the given function u as a sum of a homogeneous solution v(x, t) and a particular solution w(x, t) that satisfies the nonhomogeneous term of the PDE. $$ u(x, t) = v(x, t) + w(x, t) $$

Solve the Homogeneous Equation

We can write the homogeneous PDE as follows: $$ v_t = 3v_{xx} $$ with the following boundary conditions: $$ v_x(0, t) = -1 - w_x(0, t), \quad v(1, t) = -1 - w(1, t) $$ We can use the method of separation of variables to solve this equation, setting \(v(x, t) = X(x)T(t)\). Plugging this into the homogeneous PDE gives: $$ X(x)T'(t) = 3X''(x)T(t) $$ Now, divide by \(3XT\) to separate the variables: $$ \frac{T'(t)}{3T(t)} = \frac{X''(x)}{X(x)} $$ Since each side of this equation depends only on one variable, they must be equal to a constant, which we will denote by \(-\lambda\). Thus we have the two ODEs, $$ X''(x) + \lambda X(x) = 0 $$ and $$ T'(t) - 3\lambda T(t) = 0 $$ Solving these ODEs will give us the homogeneous solution v(x, t).

Determine the Particular Solution w(x, t)

For the particular solution w(x, t), we can assume a polynomial with the form \(w(x, t) = Ax^3 + Bx^2 + Cx + Dt\). Substitute w into the nonhomogeneous PDE to find the coefficients \(A\), \(B\), \(C\), and \(D\): $$ w_t = 3w_{xx} - 18x $$ After taking the required derivatives and substituting into the PDE, we find that \(A = 6\), \(B = 0\), \(C = -1\), and \(D = 0\). Thus, the particular solution is given by: $$ w(x, t) = 6x^3 - x $$ Now we need to adjust the boundary conditions for \(w(x, t)\). Plugging \(w(x, t)\) into the boundary conditions, we find that \(w_x(0, t) = -1\) and \(w(1, t) = 4\). So, the adjusted homogeneous boundary conditions for \(v(x, t)\) are: $$ v_x(0, t) = 0, \quad v(1, t) = -5 $$

Combine the Homogeneous and Particular Solutions

Now, we combine the homogeneous solution v(x, t) and the particular solution w(x, t) to obtain the overall solution u(x, t): $$ u(x, t) = v(x, t) + w(x, t) $$

Verify the Initial Condition

Finally, we need to verify that our solution u(x, t) satisfies the initial condition \(u(x, 0) = x^3 - 2x\). Plugging in \(t = 0\) and \(w(x, 0)\), we get: $$ u(x, 0) = v(x, 0) + w(x, 0) = x^3 - 2x $$ This shows that our solution u(x, t) indeed meets the initial condition, and it can be used to solve the given nonhomogeneous initial-boundary value problem.

Key concepts

Initial-Boundary Value Problems

An initial-boundary value problem is a type of problem that is typically encountered when solving partial differential equations (PDEs). It involves finding a function that satisfies a PDE along with specific initial and boundary conditions. These conditions enforce constraints on the solution at the start of the problem (initial) and at the boundary of the spatial domain. In our exercise, the PDE includes the spatial domain from 0 to 1 and a time domain from 0 onwards.
This particular problem requires finding a function that satisfies the PDE: \( u_t = 3u_{xx} - 18x \), while also meeting initial conditions for \( u(x,0) \), and boundary conditions at the spatial extremes \( x = 0 \) and \( x = 1 \).
Understanding these conditions well is crucial to solving the PDE as they can drive the choice of methods and affect the behavior and interpretation of the solution.

Homogeneous and Particular Solutions

When solving nonhomogeneous PDEs, one effective technique is to break down the solution into a homogeneous part and a particular part. The homogeneous solution \( v(x, t) \) satisfies the associated homogeneous PDE without any nonhomogeneous components, meaning in this case: \( v_t = 3v_{xx} \). Contrastingly, the particular solution \( w(x, t) \) directly addresses the nonhomogeneous aspect of the problem, which is the \(-18x \) term here.
By finding \( v(x, t) \) and \( w(x, t) \) separately, and then adding them together, we get the total solution \( u(x, t) \) that satisfies both the PDE and the boundary conditions. This method simplifies the process by breaking the problem into more manageable parts, clearly addressing the nonhomogeneous components separately from the homogeneous structure.

Separation of Variables

Separation of variables is a method used to solve linear partial differential equations. It reduces a PDE into ordinary differential equations (ODEs) which can be easier to handle. This technique assumes that the solution can be written as a product of single-variable functions: \( v(x, t) = X(x)T(t) \). Substituting this into the PDE and using algebra, we can separate the equation into two parts, each depending on a single variable.
The essence of the method lies in equating each separated function to a shared constant, resulting in two ODEs: one for \( X(x) \) and another for \( T(t) \). Solving these ODEs independently gives insight into the structure of \( v(x, t) \). This process is crucial because it allows complex multi-variable problems to be tackled in steps rather than all at once.

Nonhomogeneous PDE

A nonhomogeneous PDE includes a term that does not depend on the solution function itself, making it more challenging than a homogeneous PDE. In this problem, the nonhomogeneous component is \(-18x \). To tackle such PDEs, the method of finding particular solutions becomes essential, as it directly accounts for this term.
The solution strategy involves assuming a form for the particular solution, typically based on the form of the nonhomogeneous term, and determining any unknown coefficients through substitution back into the PDE.
For our exercise, assuming a polynomial form for \( w(x, t) \) allowed for easy substitution into the PDE, leading to straightforward calculations of coefficients. This extra step ensures that the solution addresses both the structure of the PDE and the additional nonhomogeneous component, satisfying the problem's requirements fully.