Question

Show that the assumptions of Theorem 13.1 .3 imply that the unique solution of $$ L y=F, \quad B_{1}(y)=k_{1}, \quad B_{2}(y)=f_{2} $$ is $$ y=\int_{a}^{b} G(x, t) F(t) d t+\frac{k_{2}}{B_{2}}\left(y_{1}\right) y_{1}+\frac{k_{1}}{B_{1}\left(y_{2}\right)} y_{2} $$

Short answer

Answer: The unique solution of the given boundary value problem is given by the expression: \(y(x)=\int_{a}^{b} G(x, t) F(t) d t+\frac{k_2}{B_{2}(y_1)}y_1(x)+\frac{k_1}{B_{1}(y_2)}y_2(x)\)

Step-by-step solution

Recall the concept and properties of Green's function

Green's function, denoted as G(x, t), is the solution of the homogeneous BVP for the operator L, i.e. $$ L G(x, t) = \delta(x - t) $$ where \(\delta(x - t)\) is the Dirac delta function. The function G(x, t) also satisfies the homogeneous boundary conditions, i.e. \(B_1(G(x,t))=0\) and \(B_2(G(x,t))=0\). It can be used to construct the solution of the inhomogeneous BVP due to the linearity of the operator L and the boundary conditions.

Obtain the complementary functions \(y_1\) and \(y_2\) satisfying the boundary conditions

Find the complementary functions \(y_1\) and \(y_2\), which are solutions to the inhomogeneous BVP with single non-homogeneous boundary conditions. For these functions, we have $$ Ly_1 = 0, \quad B_1(y_1) = 0, \quad B_2(y_1) = k_2 \\ Ly_2 = 0, \quad B_1(y_2) = k_1, \quad B_2(y_2) = 0 $$ These functions will be used in the construction of the unique solution of the given inhomogeneous BVP.

Construct the expression for the unique solution

Write down the proposed expression for the unique solution y(x): $$ y(x)=\int_{a}^{b} G(x, t) F(t) d t+\frac{k_2}{B_{2}(y_1)}y_1(x)+\frac{k_1}{B_{1}(y_2)}y_2(x) $$

Verify that the constructed expression satisfies the inhomogeneous BVP

Apply L and the boundary conditions on the proposed solution y(x) to verify that it indeed satisfies the stated inhomogeneous BVP. 1. Apply L on y(x) and show that \(L y = F\): Substitute the expression of y(x) into L, which is linear, and use the properties of Green's function and complementary functions derived in Steps 1-2. $$ L y(x) = L \left[ \int_{a}^{b} G(x, t) F(t) d t +\frac{k_2}{B_{2}(y_1)}y_1(x)+\frac{k_1}{B_{1}(y_2)}y_2(x) \right] $$ By the linearity of the operator L, $$ L y(x) = \int_{a}^{b}F(t) L G(x,t) dt + \frac{k_2}{B_{2}(y_1)} L y_1 + \frac{k_1}{B_{1}(y_2)} L y_2 $$ Using the properties of Green's function and complementary functions, $$ L y(x) = \int_{a}^{b}F(t) \delta(x - t) dt + 0 + 0 = F(x) $$ 2. Apply the boundary conditions on y(x) and show that \(B_1(y) = k_1\) and \(B_2(y) = k_2\): By properties of the Green's function and complementary functions, it is clear that the first term in our expression for y(x) will not contribute to the boundary conditions. Therefore, we only need to consider the last two terms when applying the boundary conditions. $$ B_1(y) = \frac{k_2}{B_{2}(y_1)}B_1(y_1) + \frac{k_1}{B_{1}(y_2)}B_1(y_2) = 0 + k_1 = k_1 \\ B_2(y) = \frac{k_2}{B_{2}(y_1)}B_2(y_1) + \frac{k_1}{B_{1}(y_2)}B_2(y_2) = k_2 + 0 = k_2 $$ Since the expression for y(x) satisfies the inhomogeneous BVP, it is the unique solution $$\boxed{ y(x)=\int_{a}^{b} G(x, t) F(t) d t+\frac{k_2}{B_{2}(y_1)}y_1(x)+\frac{k_1}{B_{1}(y_2)}y_2(x)}.$$

Key concepts

Boundary Value Problems

Boundary Value Problems (BVPs) are crucial in the study of differential equations, as they describe scenarios where solutions are not only determined by the equation itself but also by specific conditions at the endpoints of the interval. In simpler words, you solve for a function that satisfies both a differential equation and fixed conditions at the boundaries. This might sound daunting, but think of it like solving a puzzle where not only do the pieces have to fit together perfectly, but the edges also have to match a predefined frame.

In mathematics, BVPs arise in various physical contexts, from heat distribution along a rod to vibrations in a string. They help in modeling the behavior of systems under fixed constraints. The conditions on the boundaries can hugely influence the outcome, much like how starting and ending points can determine a journey's path.

• Boundary conditions: These are constraints that specify the values of the solution at the boundaries of the domain.
• Types include Dirichlet (fixed value at boundary), Neumann (fixed derivative at boundary), or mixed types.

The goal is to find a function that meets both the differential equation and these boundary conditions perfectly.

Differential Operators

Differential operators are mathematical tools used to describe and analyze the behavior of functions. They are essential in the formulation of differential equations, which model dynamic changes. Imagine them as operators that act on functions, similar to how arithmetic operations like addition or multiplication act on numbers.

One common differential operator is the derivative (\( \frac{d}{dx} \)), which measures rates of change. In Boundary Value Problems, a more general differential operator \( L \) is often used to represent a composed operation on functions. For example, \( L \) could be the composition of several differentiations and multiplications by variable functions, as seen in physical processes.

• Linear differential operators: These operators maintain the property of linearity over addition and scalar multiplication.
• They are the building blocks of linear differential equations and play a pivotal role in BVPs.

Understanding differential operators allows you to dissect and solve differential equations, providing a pathway to finding the behavior of complex systems.

Linearity of Differential Equations

Linearity is a defining characteristic of many differential equations, especially in Boundary Value Problems. This property means that if a particular solution of the differential equation scales or sums with another solution, the outcome is still a solution. Imagine this as stacking LEGO bricks: the result is always a construction as long as you follow the rules (i.e., the boundary conditions and the equation itself).

When dealing with linear differential equations, several advantages arise:

• Superposition Principle: Two solutions can be added or scaled, yielding new solutions.
• Predictability: Solutions maintain form, simplifying the resolution of complex equations.

The exercise's use of Green's Functions relies on this linearity. They help in constructing solutions by breaking down complex inhomogeneous problems into manageable steps, leveraging linearity to form complete solutions from basic components.

Thus, understanding linearity not only aids in solving equations more easily but also allows for more general solutions in engineering and physics applications.