Question

Using a procedure similar to that described in the text for SOLDOs, show that for the FOLDO \(\mathbf{L}_{x}=p_{1} d / d x+p_{0}\) (a) the indefinite GF is $$\begin{array}{c}G(x, y) \equiv \frac{\mu(y)}{p_{1}(y) w(y)}\left[\frac{\theta(x-y)}{\mu(x)}\right]+C(y) \\ \text { where } \mu(x)=\exp \left[\int^{x} \frac{p_{0}(t)}{p_{1}(t)} d t\right] \end{array}$$ (b) and the GF itself is discontinuous at \(x=y\) with $$\lim _{\epsilon \rightarrow 0}[G(y+\epsilon, y)-G(y-\epsilon, y)]=\frac{1}{p_{1}(y) w(y)}$$ (c) For the homogeneous \(\mathrm{BC}\) $$\mathbf{R}[u] \equiv \alpha_{1} u(a)+\alpha_{2} u^{\prime}(a)+\beta_{1} u(b)+\beta_{2} u^{\prime}(b)=0$$ construct \(G(x, y)\) and show that $$G(x, y)=\frac{1}{p_{1}(y) w(y) v(y)} v(x) \theta(x-y)+C(y) v(x),$$ where \(v(x)\) is any solution to the homogeneous \(\operatorname{DE} \mathbf{L}_{x}[u]=0\) and $$C(y)=\frac{\beta_{1} v(b)+\beta_{2} v^{\prime}(b)}{\mathbf{R}[v] p_{1}(y) w(y) v(y)}, \quad \text { with } \mathbf{R}[v] \neq 0$$ (d) Show directly that \(\mathbf{L}_{x}[G]=\delta(x-y) / w(x)\).

Short answer

The Green's function is given by functions dependent on a first-order linear differential operator, and is discontinuous at \(x = y\). When specific conditions are satisfied, applying the operator on the Green's function results in a delta function divided by the weight function.

Step-by-step solution

Calculate the indefinite Green's function

Use the given indefinite Green's function formula, plug in the functions for \(p_1(x)\), \(p_0(x)\) and \(\mu(x)\) to obtain the indefinite Green's function for specific \(x\) and \(y\). Here, \(\mu(x)=\exp \left[\int^{x} \frac{p_{0}(t)}{p_{1}(t)} d t\right]\).

Show the discontinuity of Green's function at x=y

Apply limit laws for the case when \(x\) goes to \(y\). Thus, let \(x = y + \epsilon\) and \(y = y - \epsilon\), and apply the limit as \(\epsilon\) goes to \(0\) to the Green's function for the operator in part (a). After simplification, you should reach the formula \(\frac{1}{p_{1}(y) w(y)}\).

Find the specific Green's function for the homogeneous BC

The specific Green's function \(G(x, y)\) for the homogeneous boundary condition is given as a function of \(v(x)\); substitute \(v(x) = \mu(x)\) into the definition. Consider the definition of \(C(y)\) in order to insert it in the expression for \(G(x, y)\). Proceed similarly to how you handle part (a). Check the equation you get for the homogeneous boundary condition.

Show that operator applied to Green's function equals delta function over weight function

Apply the operator \(\mathbf{L}_x\) to \(G(x,y)\) and rearrange the equation to get the delta function divided by the weight function. You must use the properties of the delta function and the specific form you found for \(G(x,y)\) in the previous step.

Key concepts

Differential Equations

Differential equations are equations involving derivatives of a function or functions. A common form is the first-order linear differential operator (FOLDO), such as \(\mathbf{L}_{x}=p_{1} \frac{d}{dx}+p_{0}\), which acts on a function \(u(x)\). Here, \(p_1(x)\) and \(p_0(x)\) are coefficients that can vary with \(x\). These equations describe a relationship between a function and its rate of change, effectively modeling various physical phenomena.For instance, in the problem at hand, we are solving for a Green's function, which assists in finding solutions to such differential equations. Green's function essentially converts a differential equation into an algebraic one, allowing for easier solutions in systems with linearity properties. The solution relies on constructing an appropriate Green's function that satisfies the given differential equation and corresponding boundary conditions.

Boundary Conditions

Boundary conditions (BCs) are additional constraints in a differential equation problem. They specify the behavior of the solution at the boundaries of the domain. In the context of this exercise, the homogeneous boundary conditions are given as \[ \mathbf{R}[u] \equiv \alpha_{1} u(a)+\alpha_{2} u'(a)+\beta_{1} u(b)+\beta_{2} u'(b)=0,\] where \(a\) and \(b\) represent boundary points.These conditions are crucial because they ensure the uniqueness of the solution to a differential equation. When dealing with Green's functions, boundary conditions are used to find the specific form of the function \(G(x,y)\), which satisfies \(\mathbf{L}_{x}[G]=\delta(x-y)/w(x)\). Thus, properly accounting for boundary conditions helps in constructing solutions that are physically meaningful.To find the appropriate Green's function, we substitute specific solutions into the boundary condition formula, such as substituting \(v(x) = \mu(x)\), thereby customizing \(G(x,y)\) to meet the BCs.

Discontinuity

Discontinuity in mathematical functions refers to points at which a function is not continuous. For Green's functions, this discontinuity typically occurs at the point \(x = y\). In this problem, the discontinuity is calculated by taking limits around \(x = y\), leading to the expression \[ \lim_{\epsilon \to 0}[G(y+\epsilon, y)-G(y-\epsilon, y)]=\frac{1}{p_{1}(y) w(y)}.\]This discontinuity is essential because it is a hallmark of Green's functions, which are used to handle systems with impulse-like inputs. The presence of a delta function in differential equations indicates the need for such a discontinuity at specific points.Through understanding and calculating this discontinuity, one can ascertain the behavior of Green's functions across the domain, providing critical insight into their roles within solutions to differential equations.

Operator Theory

Operator theory in mathematics deals with the study of operators, which are mappings between function spaces. In differential equations, operators like \(\mathbf{L}_{x}=p_{1} \frac{d}{dx}+p_{0}\) define how functions are manipulated within those equations.In this context, operator theory underpins how we apply the differential operator to Green's functions. By acting on \(G(x,y)\) with \(\mathbf{L}_{x}\), we aim to recover the delta function \(\delta(x-y)/w(x)\). This demonstrates the property of the Green's function acting as an inverse operator, converting \(\mathbf{L}_{x}[u] = g(x)\) into an expression involving only \(g(y)\).Understanding operator theory helps practitioners grasp why Green's functions are particular solutions to differential equations—they nullify all left-hand side terms except at specific points \(x = y\). By mastering these concepts, one gains the ability to transition differential problems into more manageable algebraic forms, streamline calculations, and provide deeper insights into the behavior of physical systems. Thus, operator theory is a critical mathematical tool in solving linear systems and offers a powerful way to approach differential equations.