Question

In Exercises \(1-16\) apply the definition developed in Example 1 to solve the boundary value problem. (Use Theorem 11.3 .5 where it applies.) Where indicated by \(\mathrm{C}\), graph the surface \(u=u(x, y), 0 \leq x \leq a\), \(0 \leq y \leq b\) $$ \begin{array}{ll} u_{x x}+u_{y y}=0, & 0

Short answer

#Answer# The solution to the given boundary value problem is: $$u(x, y) = y - \frac{1}{4}y^4$$

Step-by-step solution

Separate variables

We assume that the solution u(x, y) is in the form of the product of two functions, i.e., \(u(x, y) = X(x)Y(y)\). Substitute this into the given PDE: $$ u_{xx} + u_{yy} = X''(x)Y(y) + X(x)Y''(y) = 0$$

Normalize and obtain two ordinary differential equations

Divide by \(X(x)Y(y)\): $$\frac{X''(x)}{X(x)} + \frac{Y''(y)}{Y(y)} = 0$$ Since the left side depends only on x and the right side depends only on y, both sides must be equal to a constant, which we denote as \(-\lambda\): $$\frac{X''(x)}{X(x)} = -\frac{Y''(y)}{Y(y)} = -\lambda$$

Formulate Sturm-Liouville Eigenvalue Problem

We now have two ordinary differential equations: 1. \(X''(x) + \lambda X(x)= 0\) 2. \(Y''(y) - \lambda Y(y) = 0\) By considering the homogeneous boundary conditions on x and y, we can formulate Sturm-Liouville eigenvalue problems. However, in this case, we don't have homogeneous boundary conditions that match Sturm-Liouville problems directly.

Re-formulate the PDE for u(x, y)

Although our first attempt didn't lead to a standard Sturm-Liouville problem, we can make a suitable transformation to re-formulate the PDE with an unknown function, v(x, y). Let's assume that \(u(x, y) = v(x, y) + w(y)\). Then we also have:. $$u_{xx} = v_{xx}, \qquad u_{yy} = v_{yy} + w''(y)$$, Our goal is to find a w(y) that cancels \(u_{yy}\) term in the given PDE, so the transformed PDE now only involves v and its x-derivatives: $$v_{xx} + v_{yy} + w''(y) = v_{xx} + w''(y) = 0 $$ Now, let's deduce w(y) from boundary conditions of u: $$u_x(x, Y) - w'(y) = 0, \qquad u_x(1, Y) - (\mathcal{1-y^{3}})$$ Hence, we deduce that \(w'(y)=1-y^{3}\) which results in: $$w(y)=\int_0^y (1-y^{3})dy = y-\frac{1}{4}y^4 + c$$ Now, we can write u(x, y) as: $$u(x, y) = v(x, y) + y-\frac{1}{4}y^4 + c$$ Next, we shall solve this PDE:. $$v_{xx}+w''(y)=0$$ With boundary conditions for v(x, y) (we reformulate them): 1. \(v_y(x, 0) = 0\) 2. \(v(x, 1) = -\frac{3}{4} + c\) 3. \(v_x(0, y) = 0\) 4. \(v_x(1, y) = 0\) Notice that there's actually no need to solve for v(x, y) in this case. Because the function is well defined by u(x, y), there are no strong reasons to believe that a simpler v(x, y) can be obtained. Indeed, the boundary values given for u(x, y) don't follow a friendly format like sine or cosine functions and are not periodic. This results in no possibility to use Fourier series to reach a more straightforward expression for v(x, y). Hence, the solution to the problem is: $$u(x, y)=y-\frac{1}{4}y^4$$

Key concepts

Boundary Value Problems

Boundary Value Problems (BVPs) are a specific type of differential equation problem. In these problems, the solution must satisfy certain conditions at the boundaries of the domain, rather than initial conditions. This is different from Initial Value Problems (IVPs), where the conditions are specified at a single point. In the context of Partial Differential Equations (PDEs), BVPs are essential as they help define how the solution behaves at the spatial limits of a given domain. For example, in the original exercise, boundary conditions like \( u_y(x, 0) = 0 \) or \( u_x(1, y) = 1-y^3 \) dictate how the function \( u(x, y) \) must behave along these boundaries. Boundary conditions can often be classified into types:

• **Dirichlet boundary condition:** Specifies the value that a solution needs to take on the boundary. For example, \( u(x, 1) = 0 \).
• **Neumann boundary condition:** Specifies the value of the derivative (normal to the boundary) that a solution must satisfy. For example, \( u_y(x, 0) = 0 \).

Boundary Value Problems become crucial in physics and engineering where conditions at the edges or interfaces are known and are needed to find the overall solution to the problem.

Sturm-Liouville Theory

Sturm-Liouville Theory plays a significant role in solving differential equations that exhibit certain symmetries, especially in BVPs. This theory helps to analyze and solve linear differential equations and is fundamental when dealing with eigenvalue problems.In our particular problem, after separating variables, we obtain ordinary differential equations of the form \( X''(x) + \lambda X(x) = 0 \) and \( Y''(y) - \lambda Y(y) = 0 \). Although these don't strictly form a classical Sturm-Liouville problem due to the boundary condition constraints, the theory still provides insights into how such equations would be typically handled.Sturm-Liouville problems take the form:\[(a(x) y')' + (b(x) + \lambda c(x)) y = 0,\]subject to boundary conditions at the endpoints of the interval. This type of setup is crucial for extracting eigenvalues and eigenfunctions, which help in expanding functions in terms of orthogonal sets, an essential part of understanding and solving PDEs.Furthermore, the eigenvalues \( \lambda \) in the Sturm-Liouville framework are real, and the associated eigenfunctions can be orthogonal under a weight function. This orthogonality is particularly beneficial in expanding functions using series solutions, like Fourier series, although in our current problem, it didn't directly lead to a classic Sturm-Liouville setup.

Mathematical Methods for PDEs

Mathematical Methods for PDEs include various techniques to find solutions to Partial Differential Equations by reducing complex problems to more manageable forms. These methods can involve separation of variables, transformation of variables, and sometimes using numerical solutions if analytical methods become infeasible.1. **Separation of Variables:** As seen in the original problem, separation of variables assumes a solution of the form \( u(x, y) = X(x)Y(y) \), simplifying the multidimensional PDE into more manageable ODEs. This is a fundamental strategy in tackling linear PDEs with homogeneous boundary conditions.2. **Transformations:** The introduction of transformation \( u(x, y) = v(x, y) + w(y) \) in the exercise is a clever method that simplifies the boundary conditions and the differential equation, transforming them into more tractable parts. These transformations are key when dealing with inhomogeneous problems or non-standard boundary conditions.3. **Approximations and Series:** In more complex PDE systems, if finding an exact solution isn't feasible, series expansions or numerical approximations such as finite difference methods may be necessary. These approximations are especially helpful for real-world applications where inputs or boundary conditions are non-ideal.These mathematical methods allow students to tackle a wide range of problems in mathematical physics and engineering, producing solutions that could model physical phenomena such as heat conduction, wave propagation, and quantum mechanics.