Question

Solve for \(\phi(x)\) the integral equation $$ \phi(x)=f(x)+\lambda \int_{0}^{1}\left[\left(\frac{x}{y}\right)^{n}+\left(\frac{y}{x}\right)^{n}\right] \phi(y) d y $$ where \(f(x)\) is bounded for \(0

Short answer

For \( \lambda = 1 \), the solution is \( \phi(x) = \sum_{m} F_m x^{m} \). For other \( \lambda \) values causing no solutions, \( \frac{1}{\lambda} = \frac{1}{m-n} + \frac{1}{m+n} \).

Step-by-step solution

Understanding the Integral Equation

We are given the integral equation: $$\phi(x)=f(x)+\lambda\int_{0}^{1}\left[\left(\frac{x}{y}\right)^{n}+\left(\frac{y}{x}\right)^{n}\right]\phi(y) dy$$We need to solve this equation for \( \phi(x) \). Note that \( f(x) \) is a bounded function for \( 0 < x < 1 \) and \( -\frac{1}{2}<n<\frac{1}{2} \).

Simplify the Integral Kernel

The integral kernel can be simplified as: $$ K(x, y) = \left( \frac{x}{y} \right)^{n} + \left( \frac{y}{x} \right)^{n} $$

Substitute Kernel Back into the Integral Equation

Substitute \( K(x, y) \) back into the original integral equation, yielding: $$ \phi(x) = f(x) + \lambda \int_{0}^{1} \left[ \left( \frac{x}{y} \right)^{n} + \left( \frac{y}{x} \right)^{n} \right] \phi(y) dy $$

Solution for Case \( \lambda = 1 \)

For \( \lambda = 1 \), rewrite the equation: $$ \phi(x) = f(x) + \int_{0}^{1} \left[ \left( \frac{x}{y} \right)^{n} + \left( \frac{y}{x} \right)^{n} \right] \phi(y) dy $$ Assume a solution of the form: $$ \phi(x) = \sum_{m} c_{m} x^{m} $$ Substitute this assumed solution back into the integral equation and match coefficients to solve for \( c_{m} \).

Coefficients for \( c_m \)

Substituting \( \phi(x) = \sum_{m} c_{m} x^{m} \) into the equation: $$ \phi(x) = \sum_{m} c_{m} x^{m} = f(x) + \int_{0}^{1} \left[ \left( \frac{x}{y} \right)^{n} + \left( \frac{y}{x} \right)^{n} \right] \sum_{m} c_{m} y^{m} dy $$ Separate terms: $$ \sum_{m} c_{m} x^{m} = f(x) + \int_{0}^{1} \sum_{m} c_{m} \left[ x^{n} y^{-n} + x^{-n} y^{n} \right] y^{m} dy $$ Compute integrals yielding: $$ \sum_{m} c_{m} x^{m} = f(x) + \sum_{m} c_{m} \left[ x^{n} \int_{0}^{1} y^{m-n-1} dy + x^{-n} \int_{0}^{1} y^{m+n-1} dy \right] $$

Evaluate the Integrals

Evaluate the integrals: $$ \int_{0}^{1} y^{m-n-1} dy = \frac{1}{m-n}, \quad \int_{0}^{1} y^{m+n-1} dy = \frac{1}{m+n} $$ Substituting back into the equation: $$ \sum_{m} c_{m} x^{m} = f(x) + \sum_{m} c_{m} \left[ x^{n} \frac{1}{m-n} + x^{-n} \frac{1}{m+n} \right] $$

Matching Coefficients

Match the coefficients of \( x^{m} \). By analogy, $$ c_{m} = F_{m} = \int_{0}^{1} f(y) y^{m} dy $$ \( \therefore \phi(x) = \sum_{m} F_{m} x^{m} \) when \( \lambda = 1 \).

Condition for No Solution Values of \( \lambda \)

For cases where there are no solutions, the integral equation simplifies. If \( \lambda \) causes the terms \( 1 - \lambda \left[ \frac{1}{m-n} + \frac{1}{m+n} \right] = 0 \), then the integral's value must be zero unless specific values are taken: $$ \frac{1}{\lambda} = \frac{1}{m-n} + \frac{1}{m+n} $$ Causes \( \lambda = \frac{(m-n)(m+n)}{2m} \). For solutions, \( F_{\text{boundary conditions}} = c_{s} \).

Key concepts

integral kernel

In an integral equation, the integral kernel is a function that describes the relationship between the variables in the integral. Here, it takes the form:

\[ K(x, y) = \left( \frac{x}{y} \right)^{n} + \left( \frac{y}{x} \right)^{n} \]

This kernel captures how the value of \( \phi(y) \) affects \( \phi(x) \). It's expressed in terms of \( x \) and \( y \) raised to the power of \( n \) or its reciprocal.

Simplifying the kernel by expressing it in terms of \( x \) and \( y \) allows us to better understand and manipulate the integral equation.

bounded functions

A function \(f(x)\) is bounded if there's a limit to its values within a specified interval. In the given integral equation:

\[ \phi(x) = f(x) + λ \int_0^1 K(x, y) \phi(y) dy \]

It's specified that \( f(x) \) is bounded for the interval \( 0 < x < 1 \). This means that \( f(x) \) doesn't grow arbitrarily large or small within this range. This bounded nature of \( f(x) \) simplifies the process of finding \( \phi(x) \). When solving, this characteristic helps in ensuring the solution doesn't diverge.

coefficient matching

Coefficient matching is a method used to find the constants in a solution when the function is assumed to be a series. Assuming:

\[ \phi(x) = \sum_m c_m x^m \]

Substituting this back into the integral equation, and equating the coefficients of corresponding powers of \( x \) on both sides helps determine the values of \( c_m \).

By integrating and comparing coefficients, we get:

\[ c_{m} = F_{m} = \int_{0}^{1} f(y) y^{m} dy \]

This ties the coefficients of the solution to the coefficients obtained from the integral involving \( f(x) \). This method is crucial for solving the integral equation and ensures we correctly express \( \phi(x) \).

evaluating integrals

Evaluating integrals is essential in solving integral equations. Here, we need to handle two integrals separately: the ones involving terms \( y^{m-n-1} \) and \( y^{m+n-1} \).

For example:

\[ \int_{0}^{1} y^{m-n-1} dy = \frac{1}{m-n}, \ \int_{0}^{1} y^{m+n-1} dy = \frac{1}{m+n} \]

These integrals are simplified by using the properties of definite integrals of power functions. The results are then substituted back into the integral equation to match coefficients. This process reduces the complexity of the original problem, breaking it down into simpler parts that we can manage step by step.