Problem 26 - Advanced Engineering Mathematics

Question

$$\begin{aligned} &\frac{\partial^{2} u}{\partial x^{2}}=-24 x y, \frac{\partial^{2} u}{\partial y^{2}}=24 x y \text { gives } \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0 . \text { Thus } u \text { is harmonic. Now } \frac{\partial u}{\partial x}=4 y^{3}-12 x^{2} y+1=\frac{\partial v}{\partial y} \text { implies }\\\ &v=y^{4}-6 x^{2} y^{2}+y+h(x), \frac{\partial u}{\partial y}=12 x y^{2}-4 x^{3}=-\frac{\partial v}{\partial x}=12 x y^{2}-h^{\prime}(x) \text { implies } h^{\prime}(x)=4 x^{3} \quad \text { or } \quad h(x)=x^{4}+C\\\ &\text { Therefore } f(z)=4 x y^{3}-4 x^{3} y+x+i\left(y^{4}-6 x^{2} y^{2}+y+x^{4}+C\right) \end{aligned}$$

Step-by-Step Solution

Answer

The complex function is \( f(z) = 4xy^3 - 4x^3y + x + i(y^4 - 6x^2y^2 + y + x^4 + C) \).

Step 1: Verify Harmonic Condition

To begin, check that the given function is indeed harmonic. We have two second partial derivatives given: \( \frac{\partial^{2} u}{\partial x^{2}} = -24xy \) and \( \frac{\partial^{2} u}{\partial y^{2}} = 24xy \). Summing these gives: \( \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} = -24xy + 24xy = 0 \). This confirms that \( u \) is harmonic.

Step 2: Find the Conjugate Function

We are given the condition \( \frac{\partial u}{\partial x} = 4y^3 - 12x^2y + 1 = \frac{\partial v}{\partial y} \). Integrating \( v_y = 4y^3 - 12x^2y + 1 \) with respect to \( y \), we find \( v = y^4 - 6x^2y^2 + y + h(x) \), where \( h(x) \) is an arbitrary function of \( x \) to be determined.

Step 3: Differentiate and Match Terms

We now have \( \frac{\partial u}{\partial y} = 12xy^2 - 4x^3 \). From the given conditions, this also equals \( -\frac{\partial v}{\partial x} = 12xy^2 - h'(x) \). Comparing, we find \( h'(x) = 4x^3 \). Integrating this, we get \( h(x) = x^4 + C \), where C is a constant.

Step 4: Construct the Complex Function

Using the results from the previous steps, we construct the complex function \( f(z) = u + iv \). We have \( u = 4xy^3 - 4x^3y + x \) and \( v = y^4 - 6x^2y^2 + y + x^4 + C \). Therefore, \( f(z) = \left(4xy^3 - 4x^3y + x\right) + i\left(y^4 - 6x^2y^2 + y + x^4 + C\right) \).

Key Concepts

Partial Differential Equations

Partial differential equations (PDEs) are equations that involve rates of change with respect to continuous variables. These equations are crucial in mathematical physics and engineering because they describe various physical phenomena such as heat, sound, electricity, and more. In this exercise, we focus on a particular kind of PDE—harmonic functions.
Harmonic functions are solutions to Laplace's equation, which is a type of second-order PDE. Laplace's equation in two dimensions is given by:

• \( \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} = 0 \)

This equation implies that the sum of second derivatives equals zero, a condition satisfied in our original exercise, confirming that the function \( u \) is harmonic.
The presence of harmonic functions often simplifies complex problems, as they are infinitely differentiable and have certain symmetry properties. Understanding how to verify and work with harmonic functions in partial differential equations can significantly enhance one’s ability to model and solve real-world problems.

Complex Functions

Complex functions extend the concept of real-valued functions to the complex plane. In this exercise, we construct a complex function \( f(z) \), where \( z = x + iy \). The function \( f(z) \) consists of a real part \( u(x, y) \) and an imaginary part \( iv(x, y) \).
The process of constructing a complex function from a given real part involves finding a conjugate function \( v(x, y) \). This involves using the Cauchy-Riemann equations, which are conditions for a function to be holomorphic (complex differentiable).
In this exercise:

• The real part \( u \) is harmonic, verifying that \( abla^2 u = 0 \).
• Finding \( v \) involves integrating and matching terms with the help of the partial derivatives, adhering to the conditions \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \).
• Finally, the constructed complex function \( f(z) = u + iv \) combines these parts for a complete complex expression.

Creating complex functions by this method provides deeper insights into their analytical properties and is important in fields such as fluid mechanics and electrostatics.

Integral Calculus

Integral calculus deals with the accumulation of quantities and the spaces under and between curves, often used for finding areas, volumes, central points, and many useful things.
In the exercise, integral calculus is applied to find the arbitrary function \( h(x) \) when constructing \( v(x, y) \) as an integral component of the complex function. Specifically:

• Given \( h'(x) = 4x^3 \), we integrated this derivative to find \( h(x) \).

Integration yields:

• \( h(x) = \int 4x^3 \, dx = x^4 + C \)

Where \( C \) is a constant of integration. This constant can be adjusted based on boundary conditions and specific applications.
Integrating, whether entirely or partially, facilitates the transition from rates of change (derivatives) back to the function itself. Mastery of integral calculus is essential in mathematical modeling, particularly in complex analysis where integrals play a crucial role.