Lecture 9
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REGULAR PERTURBATION THEORY: This lecture introduces the formal approximation technique of perturbation theory, highlighting its broad use in initial and boundary value problems.
Lecture 10
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THE POINCARE-LINDSTED METHOD: This lecture introduces the Poincare-Lindsted perturbation method which overcomes problems with regular perturbation theory for oscillatory problems.
Lecture 11
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THE FORCED DUFFING OSCILLATOR: The Poincare-Lindsted method is applied to the forced Duffing oscillator, highlighting the power of perturbation theory to extract the underlying physics.
Lecture 12
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MULTIPLE-SCALE EXPANSIONS: This lecture introduces the formal approximation technique of perturbation theory using multiple scale theory whereby fast and slow scales are treated as independent. This is a more flexible framework than Poincare-Lindsted and regular perturbation theory.
Lecture 13
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THE VAN DER POL OSCILLATOR: This lecture uses multiple scale perturbation theory to characterise the canonical van der Pol oscillator, showing that the method can capture slow-scale transients.
Lecture 14
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BOUNDARY LAYER THEORY: This lecture uses multiple scale perturbation theory to characterise boundary layers that arise in singular perturbation problems for BVPs.
Lecture 15
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DOMINANT BALANCE, DISTINGUISHED LIMITS, MATCHED ASYMPTOTICS: This lecture introduces the formal approximation technique of perturbation theory for boundary layers and considers how to perform a dominant balance analysis using distinguishing limits and matched asymptotics.
Lecture 16
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INITIAL LAYERS AND LIMIT CYCLES: This lecture uses multiple scale perturbation theory to characterise initial layers and rapid transition regions that often occur in singularly perturbed initial value problems.
Lecture 17
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WKB THEORY: This lecture considers the structured perturbation technique of the WKB method, which approximates solutions using an amplitude-phase decomposition.
Lecture 18
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WKB AND TURNING POINTS: This lecture features the formal approximation technique of WKB theory and introduces the turning point problem and the asymptotic approximation of solutions to differential equations.