MAT4847 Assignment 3 Due 5 October 2026 |Partial Differential Equations I|

UNIVERSITY OF SOUTH AFRICA
College of Science, Engineering and Technology


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MAT4847: Partial Differential Equations I

Assignment 03 — Year Module, 2026

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MAT4847
Module Code:
Partial Differential Equations I
Module Name:
Question 4(a) and Question 4(b)
Questions:
03
Assignment Number:
05 October 2026
Due Date:
30
Total Marks:




Submitted in partial fulfilment of the requirements for MAT4847 — UNISA 2026

, UNISA | MAT4847 Assignment 03 — Q4(a) & Q4(b)



Question 4(a): Fourier Transform of an Indefinite Integral


Question


Question
Assume that p is a piecewise continuous absolutely integrable function on R, with
F(p)(0) = 0. If
Z x
q(x) = p(α) dα, x ∈ (−∞, ∞),
−∞

show that (explain clearly all the steps)

F(p)(α)
F(q)(α) = , α ̸= 0.
iα

(15 Marks)



Solution


Step 1: Differentiate q(x)


Given that
Z x
q(x) = p(α) dα,
−∞

we differentiate both sides with respect to x. By the Fundamental Theorem of Calculus,
since p is piecewise continuous and absolutely integrable, the derivative exists and satisfies


q ′ (x) = p(x).



Step 2: Take the Fourier Transform of Both Sides


The Fourier transform of a function f (x) is defined by

Z ∞
F(f )(α) = f (x) e−iαx dx.
−∞



Taking the Fourier transform of both sides of q ′ (x) = p(x) gives


F(q ′ )(α) = F (p)(α).



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