# Phyllis Nicolson's publications

We list below ten papers in which Phyllis Nicolson is a co-author. In several cases we give an extract from the Abstract of the paper. We also give a short extract from a paper by Daniel J Duffy on 'Crank Nicolson in Financial Engineering'.

- L Janossy and P Lockett, The Sun's Magnetic Field and the Diurnal and Seasonal Variations in Cosmic Ray Intensity,
*Proc. Royal Society of London. Series A, Mathematical and Physical Sciences***178**(972) (1941), 52-60.

**Abstract of the paper.**

The diurnal and seasonal variations in the vertical cosmic ray intensity, produced by a solar magnetic dipole field, are calculated at latitudes 0 and 45°, and compared with 'observed' data. It appears that the diurnal variation at latitude 45° can be largely accounted for by assuming the existence of a solar dipole of moment $1.1 \times 10^{34}$ gauss cm3 , a value which is consistent with observational evidence. The diurnal variation at the equator, however, cannot be explained by the hypothesis of a solar magnetic dipole field. The seasonal variation in intensity inferred by the above value for the solar dipole moment is of the same order of magnitude as the observational variation, but shows a phase discrepancy of two months.

- D R Hartree, P Nicolson, N Eyres, J Howlett and T Pearcey, Evaluation of the Solution of the Wave Equation for a Stratified Medium,
*Air Defense Research & Development Establishment, Memorandum***47**(May 1944).

- D R Hartree, P Nicolson, N Eyres, J Howlett and T Pearcey, Evaluation of the Solution of the Wave Equation for a Stratified Medium: Normalisation,
*Radar Research and Development Establishment, RRDE Report No.***279**(March 1945).

- D R Hartree, J G L Michel and P Nicolson, Practical methods for the solution of the equations of tropospheric refraction, in
*Report of the Conference 'Meteorological factors in radio wave propagation' held on 8 April 1946 at the Royal Institution, London, by The Physical Society and The Royal Meteorological Society*(1946), 127-168.

- P Nicolson,
*Three Problems in Theoretical Physics*, PhD Thesis (University of Manchester, 1946).

**From the Abstract.**

The thesis is divided into three sections each of which gives an account of a distinct piece of work. The first section is concerned with Cosmic Radiation and was done under the supervision of Dr L Janossy during 1939 and 1940. In 1940 cosmic ray work was interrupted by the war and the other two sections deal with problems arising from the war effort which were investigated under the direction of Professor D R Hartree while in the employ of the Ministry of Supply. Part I. Meson Formation and the East-West Asymmetry and Latitude effect at Sea Level. Part II. Transient behaviour in the single anode magnetron. Part III. The Solution of the partial differential equations describing the flow of heat in a medium in which internal evolution of heat occurs as a result of chemical processes.

- L Janossy and P Nicolson, Meson Formation and the Geomagnetic Effects,
*Proc. Royal Society of London. Series A, Mathematical and Physical Sciences***192**(1028) (1947), 99-114.

**From the Abstract.**

The experimental data on the cosmic-ray geomagnetic effects are used to provide information on the nature of the primary cosmic rays and on the mode of production of the meson component. The relevant arguments are first reviewed in a qualitative way and then elaborated by a quantitative analysis, which is not dependent upon any specific theory of meson production. Three main possibilities are discussed, the so-called proton, 'mixed' and soft component hypotheses (see \1 for definitions). It is concluded that the bulk of the mesons must arise from protons (or possibly other heavier positively charged particles). The analysis suggests that the average multiplicity of the process of meson production is about nine. From consideration of the asymmetry at high altitudes it seems likely that the primary radiation consists of protons and electrons (equally positive and negative) in the ratio of about one proton to four electrons.

- J Crank and Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type,
*Mathematical Proceeding Cambridge Phil. Society***43**(1) (1947), 50-67.

**Acknowledgement.**

The authors are grateful to Prof D R Hartree for many helpful suggestions. In operating the differential analyser, the authors were assisted by Messrs E C Lloyd, H W Parsons and W W J Cairns.

**Extract from the Paper.**

Hartree has suggested two methods (methods I and II below) of evaluating approximate solutions of partial differential equations in two variables, of the heat-conduction type. In the first the time derivative is replaced by a finite difference ratio, and the resulting ordinary differential equation with $x$ as independent variable is integrated numerically or mechanically. This integration is repeated for each finite step in time, and a trial and error process of solution is necessary to satisfy conditions at the two ends of the range in $x$. In Hartree's second method the range in $x$ is divided into a finite number of intervals, and the second space derivative of 0 at each point is expressed in terms of the values of 0 at that point and at the neighbouring points on each side. In this way the partial differential equation is replaced approximately by a set of first-order equations in time, two of which express the boundary conditions at $x = 0, x = 1$ to the same degree of approximation. The differential analyser has been used to obtain solutions of these equations, the integration proceeding in time. In sections 2 and 3 of the present paper, the application of Hartree's methods to equations (1) and (2) with conditions (3) is discussed, and the difficulties arising in carrying out mechanical solutions are examined.

The main purpose of this paper is to discuss a numerical method, method III below, developed by the authors in which both derivatives are replaced by finite difference ratios and the solution proceeds by finite steps in time. In a method proposed by Richardson the steps in time are overlapping which gives rise to a rapidly increasing oscillatory error. A method recently reported by the American Applied Mathematics Panel would seem liable to a similar disadvantage. In the method III below, the time steps do not overlap and an iterative process is involved at each step. In this way the oscillatory error is removed and much bigger steps in time may be used than in Richardson's treatment.

- P Nicolson and V Sarabhai, The Semi-Diurnal Variation in Cosmic Ray Intensity,
*Proceedings of the Physical Society (1926-1948)***60**(6) (1948), 509-523.

**From the Abstract.**

The experimental data on the daily variation in cosmic-ray intensity are first surveyed. It appears to be established by Rau's experiments that the energetic meson component at sea level shows a marked semi-diurnal variation which is in phase with the semi-diurnal variation shown by the barometric pressure. Several authors have suggested that this phenomenon is explicable in terms of the Pekeris theory of atmospheric oscillations. The implications of this explanation on the process of meson formation are here examined quantitatively. It is concluded that the explanation is only possible if mesons arise mainly at about 60 or 70 km. above sea level, which is highly unlikely since the corresponding cross-section for meson production would be much larger than appears plausible. In addition the experimental data on the diurnal variation are briefly discussed. This variation seems to be more complicated in character than is usually assumed and not attributable to factors such as a heliomagnetic field and fluctuations in the geomagnetic field, which are often held responsible.

- F J W Roughton, J W Legge and P Nicolson, The kinetics of haemoglobin in solution and in the red blood corpuscle., in
*F J W Roughton and J C Kendrew (eds.), The kinetics of haemoglobin in solution and in the red bloodcorpuscle in Haemoglobin, A symposium*(Butterworths Scientific Pub,. London, 1949).

- P Nicolson and F J W Roughton, A Theoretical Study of the Influence of Diffusion and Chemical Reaction Velocity on the Rate of Exchange of Carbon Monoxide and Oxygen between the Red Blood Corpuscle and the Surrounding Fluid,
*Proceedings Royal Society of London. Series B, Biological Sciences***138**(891) (1951), 241-264.

**From the Abstract.**

Numerical methods have been applied to the problem of the diffusion of CO and O2 through the red blood corpuscle membrane, accompanied by diffusion and chemical reaction of these substances in the layer of concentrated haemoglobin solution inside the corpuscle. The methods are applied to recent data on (i) the rates of CO uptake by corpuscle suspensions and haemoglobin solutions, prepared from the blood of rams and pregnant ewes, (ii) the rates of O2 egress from ram corpuscle suspensions and haemoglobin solutions. In the case of the ram blood the slower overall rates of exchange in the corpuscle suspensions, as compared with the haemoglobin solution, can be accounted for by the limiting effect of diffusion inside the corpuscle, without the necessity of attributing any limiting effect to the corpuscle membrane. In the pregnant ewe blood, on the other hand, the corpuscle membrane appears to have a definite limiting effect on the passage of CO and preliminary values of its permeability to this substance are given. It is also shown how the special methods, of computation developed in this paper can be generalised and thus made applicable to other processes in which diffusion and chemical reaction velocity are jointly involved.

**Note.**

The mathematical part of the work is mainly due to Phyllis Nicolson, and the physico-chemical and physiological parts to Francis John Worsley Roughton.

- J Crank and Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type,
*Advances in Computational Mathematics***6**(1996), 207-226.

**Note.**

This is a reprint of the paper [7] above. It is an indication of the importance of the paper that it was republished fifty years after it was written.

- Daniel J Duffy gives 'A short History of Crank Nicolson in Financial Engineering' in A Critique of the Crank Nicolson Scheme Strengths and Weaknesses for Financial Instrument Pricing,
*WILMOTT magazine*(4) (2004), 68-76. We give a short extract below.

The Crank Nicolson finite difference scheme was invented by John Crank and Phyllis Nicolson. They originally applied it to the heat equation and they approximated the solution of the heat equation on some finite grid by approximating the derivatives in space $x$ and time $t$ by finite differences. Much earlier, Richardson devised a finite difference scheme that was easy to compute but was numerically unstable and thus useless. The instability was not recognised until Crank, Nicolson and others carried out lengthy numerical calculations. In short, the Crank Nicolson method is numerically stable and it only requires the solution of a very simple system of linear equations (namely, a tridiagonal system) at every time level.

The Crank Nicolson method has become one of the most popular finite difference schemes for approximating the solution of the Black Scholes equation and its generalisations (see for example, Tavella 2000, Bhansali 1998). The method is essentially a second-order approximation to the time derivative that appears in the Black Scholes equation and this property, plus the fact that the method is stable and is easy to program makes it very appealing in practical applications. Numerous articles and publications in the financial engineering literature use Crank Nicolson as the de-facto scheme for time discretisation. Unfortunately, the method breaks down in certain situations and there are better and more robust alternatives that have been documented in the numerical analysis and computational fluid dynamics literature. To this end, we wish to discuss the shortcomings of the method and how they can be resolved.

Last Updated December 2021