Test: Laplace Equation - Engineering Mathematics MCQ

Given that,
The function f(x, y) satisfies the Laplace equation  ∇²f(x, y) = 0
on a circular domain of radius r = 1 with its center at point P with coordinates x = 0, y = 0. 
The value of this function on the circular boundary of this domain is equal to 3.
Here it is given that the value of the function is 3 for its domain, which signifies that it is a constant function whose value is 3.
So the value of the function at (0, 0) is 3.

From the basic definition of Laplace transform:

Calculation:
Given:
a = 5i

Laplacian operator in the Cartesian system is:

Laplacian operator in Cylindrical system is:

Laplacian operator in Spherical system is:

Calculation:
The Laplacian of a scalar field ψ = 2x²y - xz³

Solving the above we will get option 3 as the correct answer. 

Solutions of Laplace’s equation having continuous second-order partial derivatives is given by  and it is called a harmonic function (where Φ = any constant).
E.g. Φ = 2xy satisfies  hence it is called a harmonic function.
Conjugate of Harmonic function:
If f(z) = u + iv is an analytic function then imaginary part v is known as conjugate harmonic functions of u (But the converse is not true) and u is conjugate harmonic of (-v).

If a function ϕ(x₁, x₂) satisfies the Laplace equation,  then the function is the solution of the Laplace equation.
Calculation:
Given function is, 
Differentiating ϕ partially with respect to x₁

Again differentiating ϕ partially with respect to x₁

Similarly, differentiating ϕ partially with respect to x₂ twice, we will get,

Adding above two equations,

As the given function satisfies the Laplace equation ∇²ϕ = 0, hence it is the solution of Laplace equation.