The analytic and numerical solutions for partial differential equations are often presented as quite different. However, the analytic solution's main purpose is to validate numerical code, so it needs to be converted to a numerical solution. When this is done correctly, a simple numerical solution can be found in a few lines of code. We focus here on the solution of the canonical partial differential equations presented to students in their first course of partial differential equations. We will show the solution using separation of variables is converted in a very natural way to matrix multiplication. We will then show how this allows us to simulate the solution to high accuracy with straightforward and efficient code. The method will be extended to more complex problems, but the emphasis will be on carefully explaining the basic case that can be easily generalised once understood.
Michael Meylan obtained his B.Sc. and PhD degrees from Otago University, New Zealand, in 1991 and 1994, respectively. He held appointments at Massey University, New Zealand (1999–2003), University of Auckland, New Zealand (2003–2011) and serves currently at The University of Newcastle (Australia). His research is in wave theory, especially the coupling of elasticity and fluids. He has worked extensively in linear hydroelasticity, principally in relation to wave scattering in frozen oceans. He has also worked on applying the theory of generalised eigenfunction expansions and the singularity expansion method to problems in hydrodynamics.
Michael is recognised as a world-leading researcher in ocean wave modelling and, more broadly, in applied and industrial mathematics. I have published 145 research articles in highly regarded international journals, with an h-index of 34 and a total citation of 3244. I founded a research group in Australia in my principal research area of sea ice modelling, and with this group, I have obtained an ARC Discovery grant and two LIEF grants ($750k). I have established extensive research networks worldwide and in Australia and New Zealand. My research has had a broad and significant impact on wave prediction and climate modelling. My research leadership has been recognised by research groups seeking my collaboration and expertise. For example, groups in the United States and Canada have invited me to participate in the development of next-generation wave-ice models. I have also worked closely with a wave energy research group at the University of Plymouth led by Prof. Greaves, recipient of the Order of the British Empire for her services to engineering. I have forged strong research connections with various groups in India, and I have played a critical role in obtaining funding through the Government of India (SPARC $124k).
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