Problem 7 - Applied Partial Differential Equations

Question

For $$ L-\frac{d}{d x}\left(p \frac{d}{d x}\right)+q $$ with \(p\) and \(q\) real, carefully show that $$ \overline{L(\phi)}=L(\bar{\phi}) $$

Step-by-Step Solution

Answer

\(\overline{L(\phi)}\) is indeed equal to \(L(\bar{\phi})\). The complex conjugation of a differential operator applied to a function yields the same result as the differential operator applied to the complex conjugate of the function.

Step 1: Write down the given differential operator

The differential operator is \(L -\frac{d}{d x}\left(p \frac{d}{d x}\right) +q\)

Step 2: Apply the operator to function \(\phi\)

The operator applied to the function \(\phi\) is \(L(\phi) -\frac{d}{d x}\left(p \frac{d}{d x}\right) (\phi) + q\phi\).

Step 3: Take the complex conjugate of the function

The complex conjugate of \(L(\phi)\) is \(\overline{L(\phi)} -\overline{\frac{d}{d x}\left(p \frac{d}{d x}\right) (\phi) }+ \overline{q\phi}\). Since \(p\) and \(q\) are real, their complex conjugates are just themselves.

Step 4: Apply the operator to the complex conjugate of the function

The operator applied to the complex conjugate of the function, \(\bar{\phi}\), is \(L(\bar{\phi}) -\frac{d}{d x}\left(p \frac{d}{d x}\right) (\bar{\phi}) + q\bar{\phi}\).

Step 5: Show that the two expressions are equal

Comparing \(\overline{L(\phi)} -\overline{\frac{d}{d x}\left(p \frac{d}{d x}\right) (\phi) }+ \overline{q\phi}\) to \(L(\bar{\phi}) -\frac{d}{d x}\left(p \frac{d}{d x}\right) (\bar{\phi}) + q\bar{\phi}\), it can be seen that they are indeed equal, as we take the conjugate of the whole derivative expression in the former, and inside the derivative expression in the latter. So, \(\overline{L(\phi)}=L(\bar{\phi})\) holds true.

Key Concepts

Differential Operators

A differential operator is a type of function that involves the derivatives of a function. In the context of partial differential equations, it becomes crucial for analyzing changes with respect to multiple variables. The operator given in the exercise is \(-\frac{d}{dx}\left(p \frac{d}{d x}\right) + q\), where \(p\) and \(q\) are coefficients.

• The notation \(\frac{d}{dx}\) represents differentiation with respect to \(x\).
• Operators like this help us understand how a function behaves as its input changes, particularly when involving real-world phenomena like heat conduction or wave propagation.

The key to solving problems involving differential operators is understanding how they "act" on a function. Here, the operator is applied to a function \(\phi\), and it involves first computing the derivative, multiplying it by \(p\), and then differentiating again. This process results in transformations that provide insights into how particular processes evolve.

Complex Conjugates

Complex conjugates are a concept mainly used when dealing with complex numbers, which have a real part and an imaginary part. The conjugate of a complex number \(a + bi\) is \(a - bi\). It's a reflection across the real axis in the complex plane.

• Taking the complex conjugate of an expression doesn't change the real parts of the terms, but it negates the imaginary part.
• This property is particularly useful for ensuring certain mathematical expressions remain valid, particularly in conditions where symmetry exists, like in this exercise when \(L(\phi)\) is transformed into \(\overline{L(\phi)}\).

In this exercise, using complex conjugates helps confirm relationships between functions, showing equality between two differently approached expressions. As part of working through the differential operator, it illustrates that the transformation maintains symmetry since \(p\) and \(q\) remain unchanged being real numbers.

Real Coefficients

Real coefficients in equations refer to numbers that do not have an imaginary component. In differential equations, coefficients like \(p\) and \(q\) being real ensures certain properties about the equation itself, such as stability and symmetry.

• When coefficients are real, the equations have solutions that are often easier to interpret, akin to tangible, real-world phenomena.
• The fact that \(p\) and \(q\) are real in this exercise simplifies calculations because the complex conjugate of a real number is the number itself.

This concept is essential while comparing \(\overline{L(\phi)}\) with \(L(\bar{\phi})\) since it shortens and streamlines the computation process. The truth of \(\overline{L(\phi)} = L(\bar{\phi})\) relies on the unchanged nature of \(p\) and \(q\), as there are no imaginary parts to affect the result, highlighting consistency and predictability in the operator's behavior.