Question

use the result of Problem 32 to find the adjoint of the given differential equation. $$x^2{y}^{\prime \prime }+x{y}^{\prime }+(x^2-v^2)y=0,\text{ Bessel's equation }$$

Short answer

#Answer# The adjoint of the given Bessel's differential equation is: $$x^2(u^{″}(x)y+2u^{\prime }(x)y^{\prime }+u(x)y^{″})+x(u^{\prime }(x)y^{\prime }+u(x)y^{\prime })-x{u}^{\prime }(x)y^{\prime }+(x^2-v^2)(u(x)y)=0.$$

Step-by-step solution

Identify the coefficients

Here, we are given the equation $$x^2{y}^{″}+x{y}^{\prime }+(x^2-v^2)y=0.$$ The coefficients are $x^2$ for the second derivative term ( $y^{″}$ ), $x$ for the first derivative term ( $y^{\prime }$ ), and $(x^2-v^2)$ for the $y$ term.

Multiply by a weighting function

Now, we will multiply each term by a weighting function, which is often represented by the symbol $u(x)$, to obtain the following expression: $$u(x)x^2{y}^{″}+u(x)x{y}^{\prime }+u(x)(x^2-v^2)y=0.$$

Reverse the order of the derivatives

Next, we reverse the order of the derivatives in each term: $$x^2(u(x)y{)}^{″}+x((u(x)y{)}^{\prime }{)}^{\prime }+(x^2-v^2)(u(x)y)=0.$$

Simplify the resulting equation

Now, simplify the equation to obtain the adjoint: $$\begin{array}{rl}{x}^2(u(x)y{)}^{″}+x(u(x)y{)}^{\prime }-x{u}^{\prime }(x)y^{\prime }+(x^2-v^2)(u(x)y)& =0\\ x^2(u^{″}(x)y+2u^{\prime }(x)y^{\prime }+u(x)y^{″})+x(u^{\prime }(x)y^{\prime }+u(x)y^{\prime })-x{u}^{\prime }(x)y^{\prime }+(x^2-v^2)(u(x)y)& =0.\end{array}$$ Thus, the adjoint of the given Bessel's differential equation is: $$x^2(u^{″}(x)y+2u^{\prime }(x)y^{\prime }+u(x)y^{″})+x(u^{\prime }(x)y^{\prime }+u(x)y^{\prime })-x{u}^{\prime }(x)y^{\prime }+(x^2-v^2)(u(x)y)=0.$$

Key concepts

Differential Equations

Differential equations are equations that involve a function and its derivatives. The main goal in studying these equations is often to find the unknown function. They come in many forms and are used to model and solve problems in physics, engineering, economics, biology, and more.

Bessel's equation is a type of differential equation that frequently appears in problems with cylindrical symmetry, such as heat conduction in a cylindrical object or vibrations of a circular membrane. It has the general form:
$$x^2{y}^{″}+x{y}^{\prime }+(x^2-v^2)y=0,$$
where $y^{″}$ denotes the second derivative of $y$, $y^{\prime }$ is the first derivative, and $v$ is a constant.

To solve Bessel’s equation, one typically looks for solutions known as Bessel functions, which can be separable into specific functions of set orders. The equation itself is useful across many fields because its solutions can describe wave patterns, heat distribution, and electrical potentials, among other physical phenomena. Understanding differential equations can help predict how systems change or stabilize over time.

Adjoint Operator

The concept of an adjoint operator is vital in differential equations, especially when dealing with Bessel's equation. Adjoint operators help in understanding symmetry and conservation properties related to an equation. An operator is essentially a function that performs a specified operation on a function. The adjoint of a differential operator, on the other hand, is a derived operator that often displays symmetrical properties of the original operator.

In the context of differential equations, finding the adjoint involves manipulating the original operator generally by introducing a weighting function, rearranging the terms, and looking at the properties of the resulting operation. For Bessel’s equation, and many others, understanding the adjoint operator can simplify solving, analyzing boundary value problems or ensuring the best form for applying numerical methods on the equations.

Through the steps of identifying coefficients and reversing the order of derivatives, it becomes possible to derive the adjoint of a given equation, allowing new paths for solutions and analysis of the physical interpretation of differential equation solutions.

Bessel Functions

Bessel functions are solutions to Bessel's differential equation, and they are critical in various scientific and engineering contexts. These functions, denoted typically as $J_v(x)$ and $Y_v(x)$, where $v$ is the order of the Bessel function, represent oscillatory behavior and are especially useful for problems with circular or cylindrical symmetry.

The most commonly encountered Bessel functions are the first kind $J_v(x)$, which are finite at the origin, and the second kind $Y_v(x)$, which generally diverge at the origin. These functions exhibit oscillations similar to sine and cosine functions but are adapted to the geometry involved in the physical problems they solve.

Applications of Bessel functions are vast, including modeling wave propagation, heat conduction around cylindrical objects, electromagnetic fields, and mechanical vibrations. In solving boundary value problems, the properties of Bessel functions, like orthogonality and recurrence relations, are often leveraged, making them a powerful tool for engineers and scientists.

Understanding these functions and their properties can greatly aid in the theoretical and practical solving of complex systems modeled by differential equations.