# Partial Differential Equations Questions and Answers – Derivation and Solution of Two-dimensional Heat Equation

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This set of Fourier Analysis and Partial Differential Equations Multiple Choice Questions & Answers (MCQs) focuses on “Derivation and Solution of Two-dimensional Heat Equation”.

1. Who was the first person to develop the heat equation?
a) Joseph Fourier
b) Galileo Galilei
c) Daniel Gabriel Fahrenheit
d) Robert Boyle

Explanation: The heat equation was first put forth by Jean-Baptiste Joseph Fourier (21 March 1768 – 16 May 1830) in 1822. He was a French mathematician and physicist born in Auxerre. It was developed to describe heat flow.

2. Which of the following is not a field in which heat equation is used?
a) Probability theory
b) Histology
c) Financial Mathematics
d) Quantum Mechanics

Explanation: The heat equation is used is several fields, some of which include,

• Probability theory (to study Brownian motion)
• Financial Mathematics (for solving Black- Scholes PDE)
• Quantum Mechanics (to determine spread of wave function)

3. Under ideal assumptions, what is the two-dimensional heat equation?
a) ut = c∇2 u = c(uxx + uyy)
b) ut = c2 uxx
c) ut = c22 u = c2 (uxx + uyy)
d) ut = ∇2 u = (uxx + uyy)

Explanation: Consider a rectangular plate (thermally conductive material), with dimensions a × b. The plate is heated and then insulated.
We let u (x, y, t) = temperature of plate at position (x, y) and time t.
For a fixed t, the height of the surface z = u (x, y, t) gives the temperature of the plate at time t and position (x, y).
Under ideal assumptions (e.g. uniform density, uniform specific heat, perfect insulation, no internal heat sources etc.) one can show that u satisfies the two-dimensional heat equation,
ut = c22 u = c2 (uxx + uyy) for 0 < x < a, 0 < y < b.

4. In mathematics, an initial condition (also called a seed value), is a value of an evolving variable at some point in time designated as the initial time (t=0).
a) False
b) True

Explanation: Another definition of initial condition may be stated as, ‘any of a set of starting-point values belonging to or imposed upon the variables in an equation that has one or more arbitrary constants’

5. What is another name for heat equation?
a) Induction equation
b) Condenser equation
c) Diffusion equation
d) Solar equation

Explanation: The heat equation is also known as the diffusion equation and it describes a time-varying evolution of a function u(x, t) given its initial distribution u(x, 0).

6. Heat Equation is an example of elliptical partial differential equation.
a) True
b) False

Explanation: Jean-Baptiste Joseph Fourier (21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre who was the first person to develop heat equation. The heat equation is the prototypical example of a parabolic partial differential equation.

7. What is the half-interval method in numerical analysis is also known as?
a) Newton-Raphson method
b) Regula Falsi method
c) Taylor’s method
d) Bisection method

Explanation: The Bisection method, also known as binary chopping or half-interval method, is a starting method which is used, where applicable, for few iterations, to obtain a good initial value.

8. Which of the following represents the canonical form of a second order parabolic PDE?
a) $$\frac{∂^2 z}{∂η^2}+⋯=0$$
b) $$\frac{∂^2 z}{∂ζ∂η}+⋯=0$$
c) $$\frac{∂^2 z}{∂α^2}+\frac{∂^2 z}{∂β^2}…=0$$
d) $$\frac{∂^2 z}{∂ζ^2}+⋯=0$$

Explanation: A second order linear partial differential equation can be reduced to so-called canonical form by an appropriate change of variables ξ = ξ(x, y), η = η(x, y).

9. Which of the following is the condition for a second order partial differential equation to be hyperbolic?
a) b2-ac<0
b) b2-ac=0
c) b2-ac>0
d) b2-ac=<0

Explanation: For a second order partial differential equation to be hyperbolic, the equation should satisfy the condition, b2-ac>0.

10. What is the order of the partial differential equation, $$\frac{∂^2 z}{∂x^2}-(\frac{∂z}{∂y})^5+\frac{∂^2 z}{∂x∂y}=0$$?
a) Order-5
b) Order-1
c) Order-4
d) Order-2

Explanation: The order of an equation is defined as the highest derivative present in the equation. Hence, in the given equation, $$\frac{∂^2 z}{∂x^2}-(\frac{∂z}{∂y})^5+\frac{∂^2 z}{∂x∂y}=0$$, the order is 2.

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