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MA222 Final Exam - Introduction to Partial Differential Equations
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MCQs on Partial Differential Equations.

MA222 Final Exam - Introduction to Partial Differential Equations
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25 Questions

1. Suppose
is a sequence of real-valued functions in
and that
is a given function.Which of the following statements must be true?
I.
uniformly on
.
II.
is a continuous function.
III.
in
.
2. Which of the following is NOT equal to the Gauss-Weierstrass kernel
?
3. Which of the following statements is true?
4. Which of the following is an orthonormal set of vectors in
?
5. Let
and
denote the Fourier sine and cosine transforms of
, respectively. Which of the following holds?
I.

II.
6. Suppose that
and denote their Fourier transforms by
and
, respectively.Which of the following is the unique solution of the one dimensional wave equation
coupled with the initial conditions
?
7. Which of the following PDEs are NOT evolution equations?
I.

II.

III.
8. Let
be a function.Which of the following descriptive statements regarding its Fourier transform
are correct?
I. The Fourier transform converts differentiation in
into multiplication in
.
II. The smoothness of
is closely related to the asymptotic decay rate of its Fourier transform
?
III. The asymptotic decay rate of
determines the smoothness of its Fourier transform.
9. Which of the following functions DOES NOT have periodic boundary conditions on
?
10. Which of these formulas is equal to
, where
?
11. Assume that the Fourier sine series representation of f is
.Which of the following is the unique solution of the one dimensional heat equation
on
coupled with homogenous Neumann boundary conditions
and initial condition
?
12. Which of the following statements is true?
I. If
is uniformly convergent on
, then
is convergent in the sense of
.
II. If
is convergent in the sense of
, then
is pointwise convergent on
.
III. If
is pointwise convergent on
, then
is convergent in the sense of
.
13. Solve the following initial-value problem.
14. Determine the Fourier series representation for
on
.
15. Complete the following statement.A linear operator
is self-adjoint, if and only if
is:
16. Which of the following is an accurately-stated consequence of Gibb's phenomenon for a function
?
17. Let
be a continuous function.Which of the following is the
norm of
?
18. Suppose the real Fourier series for
on
is given by
.
Which of the following statements is false?
I. If
, then this series converges pointwise on
.
II.

III. If
, for any
, then
is an even function.
19. Which of the following statements, if any, are false?
I.
, where the inner product is for the space
.
II.
is an orthonormal set of functions in
.
III.
is an orthogonal set of functions in
.
20. Which of the following is the Fourier transform of
?
21. Let
denote the Dirac delta function concentrated at
.How many of the following statements are true?
I.

II.

III.
, for all functions

IV.
, where
is the Heaviside function
V.
, where
is a real constant
22. Solve the following initial-value problem using the method of characteristics:
23. Solve the following boundary value problem:
24. Solve the following initial-value problem using the method of characteristics:
25. Consider the Laplace equation on a circle of radius a given by

Which of the following system of ODEs in
and
arises as part of the solution process when using separation of variables?

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