### Module Details

 The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
 Title FURTHER METHODS OF APPLIED MATHEMATICS Code MATH323 Coordinator Dr GT Piliposyan Mathematical Sciences G.Piliposyan@liverpool.ac.uk Year CATS Level Semester CATS Value Session 2021-22 Level 6 FHEQ First Semester 15

### Aims

•To give an insight into some specific methods for solving important types of ordinary differential equations.

•To provide a basic understanding of the Calculus of Variations and to illustrate the techniques using simple examples in a variety of areas in mathematics and physics.

•To build on the students'' existing knowledge of partial differential equations of first and second order.

### Learning Outcomes

(LO1) After completing the module students should be able to:
- use the method of "Variation of Arbitrary Parameters" to find the solutions of some inhomogeneous ordinary differential equations.

- solve simple integral extremal problems including cases with constraints;

- classify a system of simultaneous 1st-order linear partial differential equations, and to find the Riemann invariants and general or specific solutions in appropriate cases;

- classify 2nd-order linear partial differential equations and, in appropriate cases, find general or specific solutions.

[This might involve a practical understanding of a variety of mathematics tools; e.g. conformal mapping and Fourier transforms.]

### Syllabus

Ordinary differential equations; some methods, including the variation of arbitrary constants, for solving certain types of equations.

Introduction to the Calculus of Variations for problems without and with constraints.

Simultaneous first-order linear partial differential equations; Riemann invariants.

Second-order linear partial differential equations; classification, reduction to standard forms, conformal mappings. Fourier transforms.

### Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Final Assessment Assessment Schedule (When) :First semester  1 hour time on task    50
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
class test  around 60-90 minutes    50