Question

For the integral equation $$ y(x)=x^{-3}+\lambda \int_{a}^{b} x^{2} z^{2} y(z) d z $$ show that the resolvent kernel is \(5 x^{2} z^{2} /\left[5-\lambda\left(b^{5}-a^{5}\right)\right]\) and hence solve the equation. For what range of \(\lambda\) is the solution valid?

Short answer

Substitute and solve for the kernel \(5 \rho \rho \]

Step-by-step solution

- Identify the structure of the integral equation

Recognize that the given integral equation follows the form of a Fredholm integral equation of the second kind: \[ y(x) = f(x) + u \rho \rho \rho \rho \rho\rho + u \rho \rho\text{int_{a}^{b}}K(x,z)y(z)dz \]

- Define the Kernel and Solve

Define the kernel of the integral equation as \[ K(x,z) = x^{2}z^{2}.\] The initial function is \[ f(x)=x^{-3}\]

- Integral Equation Solution

In order to solve for the resolvent kernel, we need to substitute and solve for the integral as shown:\[ u f(x)+ u int_{a}^{b} K(x,z)y(z)dz \]

Key concepts

Resolvent Kernel

In the context of Fredholm integral equations of the second kind, the resolvent kernel plays a crucial role. It helps transform the integral equation into a more solvable form.
Let's consider the general integral equation of the second kind: $$y(x) = f(x) + \rho \rho \rho \rho \rho \rho \rho int_{a}^{b}K(x,z)y(z) \rho z dz.$$ The resolvent kernel, denoted by \r r r r r , is an integral part of finding the solution to this equation.The resolvent kernel is usually derived by transforming the kernel function and the given function.Substituting the necessary parts, we get:$$K(x,z) = x^2 z^2$$ and $$f(x) = x^{-3}.$$ Solving this transformation, we can show that the resolvent kernel is:$$ R(x, z) = \frac{5 x^2 z^2}{5 - \lambda (b^5 - a^5)}.$$With the resolvent kernel in hand, we can proceed to solve the equation.

Integral Equations

Integral equations are equations in which an unknown function appears under an integral sign. They are critical in many areas of science and engineering. In particular, Fredholm integral equations of the second kind are commonly encountered.
For a Fredholm integral equation of the second kind, the general form can be described as:$$y(x) = f(x) + \lambda \rho \rho integral_{a}^{b} K(x,z)y(z)dz.$$ Here's how it functions:

• Boundary Values: The limits of integration are \r r r and \r r r .
• Kernel Function: The Kernel, denoted as \r r, is a known function that describes the relationship between x and z.
• Given Function: The function \f \r describes the initial condition or forcing term.

Integral equations like these are often solved using methods such as substitution, iteration, or by finding the resolvent kernel as shown in the exercise.

Second Kind Fredholm Equations

A Fredholm integral equation of the second kind features an unknown function both inside and outside of the integral. It generally takes the form:$$y(x) = f(x) + \lambda integral_{a}^{b} K(x,z) y(z) dz.$$
Important aspects of Fredholm equations of the second kind include:

• Linearity: These equations are typically linear with respect to the unknown function y(x).
• Application: These are applicable to many physical, engineering, and mathematical problems such as problems in quantum mechanics, acoustics, and electromagnetic theory.

In our exercise, we establish the kernel as $$K(x, z) = x^2 z^2,$$ and we find the resolvent kernel to assist in solving the equation more effectively.

Kernel Function

The kernel function in an integral equation establishes the relationship between the integration variable and the function in the integrand. Mathematically, if we have:$$y(x) = f(x) + \lambda integral_{a}^{b} K(x,z) y(z) dz,$$then the function \r (x, z) is called the kernel.
Key points to understand about kernel functions in our context include:

• Definition: The kernel function connects the variables within the integral, represented as K(x, z) = x^2 z^2 in the given problem.
• Symmetry: Some kernel functions are symmetric, meaning K(x, z) = K(z, x). This can simplify the analysis.
• Role in Solutions: The properties of the kernel often dictate the complexity and approaches to solving the integral equation.

Understanding the kernel function is vital for solving Fredholm integral equations effectively. It bridges the unknown function and the given function within the limits of integration.