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Test: Laplace Equation - Civil Engineering (CE) MCQ


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5 Questions MCQ Test Engineering Mathematics - Test: Laplace Equation

Test: Laplace Equation for Civil Engineering (CE) 2024 is part of Engineering Mathematics preparation. The Test: Laplace Equation questions and answers have been prepared according to the Civil Engineering (CE) exam syllabus.The Test: Laplace Equation MCQs are made for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Laplace Equation below.
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Test: Laplace Equation - Question 1

The function f(x, y) satisfies the Laplace equation

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.
The numerical value of f(0, 0) is:

Detailed Solution for Test: Laplace Equation - Question 1

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.

Test: Laplace Equation - Question 2

The Laplace transform of e i5t where i = √−1, is

Detailed Solution for Test: Laplace Equation - Question 2

From the basic definition of Laplace transform:

Calculation:
Given:
a = 5i

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Test: Laplace Equation - Question 3

If ψ = 2x2y - xz3, then the Laplacian of ψ is

Detailed Solution for Test: Laplace Equation - Question 3

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. 

Test: Laplace Equation - Question 4

Solutions of Laplace’s equation, which are continuous through the second derivatives are called

Detailed Solution for Test: Laplace Equation - Question 4

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).

Test: Laplace Equation - Question 5

The function  is the solution of

Detailed Solution for Test: Laplace Equation - Question 5

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.

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