# Graham W GriffithsCity, University of London

Graham W Griffiths

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56

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Introduction

Graham W Griffiths is a visiting professor at City, University of London. Graham does research in numerical analysis and climate change. His most recent publication is Analysis of cornea curvature using radial basis functions - Part I: Methodology.'

## Publications

Publications (56)

A Compendium of Partial Differential Equation Models presents numerical methods and associated computer codes in Matlab for the solution of a spectrum of models expressed as partial differential equations (PDEs), one of the mostly widely used forms of mathematics in science and engineering. The authors focus on the method of lines (MOL), a well-est...

Partial differential equations (PDEs) are a general starting point for mathematical modeling and computer-based analysis throughout all of science, engineering and applied mathematics. Computer-based methods for the numerical and analytical solution of PDEs are therefore of broad interest. In this chapter, we discuss some of the general approaches...

This book presents some of the latest numerical solutions to initial value problems and boundary value problems described by ODEs and PDEs. The author offers practical methods that can be adapted to solve a wide range of problems which are illustrated in the increasingly popular open source computer language R, which allows integration with more st...

We discuss the solution of cornea curvature using a meshless method based on radial basis functions (RBFs). A full two-dimensional nonlinear thin membrane partial di�erential equation (PDE) model is introduced and solved using the multiquadratic (MQ) and inverse multiquadratic (IMQ) RBFs. This new approach does not rely on radial symmetry or other...

An introduction to Schrodinger's equation with examples

This article provides some background to the mathematics of the neutron diffusion process and critical mass.

In part I we discussed the solution of corneal curvature using a 2D meshless method based on radial basis functions (RBFs). In Part II we use these methods to fit a full nonlinear thin membrane model to a measured data-set in order to generate a topological mathematical description of the cornea. In addition, we show how these results can lead to e...

The increasing concentration of atmospheric CO2 is now a problem of global concern. Although the consequences of atmospheric CO2 are still evolving, there is compelling evidence that the global environmental system is undergoing profound changes as seen in the recent spike of phenomena: extreme heat waves, droughts, wildfires, melting glaciers, and...

Purpose
– The purpose of this paper is to present the method of lines (MOL) solution of the stimulated Brillouin scattering (SBS) equations (a system of three first-order hyperbolic partial differential equations (PDEs)), describing the three-wave interaction resulting from a coupling between light and acoustic waves. The system has complex numbers...

The spectral and temporal evolution of distributed sensing based on stimulated Brillouin scattering (SBS) in optical fibers for severalnanosecondStokes pulses is demonstrated by using the method of lines (MOL) solution of the transient SBS equations. A superbee fluxlimiter is utilized to avoid numerical damping and dispersion that would otherwise b...

The Fisher-Kolmogorov partial differential equation (PDE) is an extension of the convection-diffusion-reaction (CDR) partial differential equation (PDE), which can be termed a mixed hyperbolic-parabolic PDE, with a linear source term and a source term of arbitrary order (the order is a parameter). A numerical solution is computed by the method of l...

Traveling wave analysis takes the form of applying a Lagrangian change of variable to a PDE to give an ODE that is easier to solve analytically, particularly if a computer algebra system such as Maple is applied to the ODE. An analytical solution to the PDE is then produced by applying the inverse of the original transformation to the ODE. This cha...

This chapter discusses the conversion of an ODE to a PDE using the concept of a traveling wave equation, then using an inverse Lagrangian transformation to obtain a PDE is applied to an ODE that leads to a PDE second order in the initial value variable. The resulting PDE has the form of the linear wave equation, so it is termed a modified wave equa...

The convection-diffusion-reaction (CDR) partial differential equation (PDE), which can be termed a mixed hyperbolic-parabolic PDE, is extended to include a second-order and a third-order source term. A numerical solution is computed by the method of lines (MOL), including detailed discussion of the Matlab routines and the numerical and graphical ou...

The Burgers-Fisher partial differential equation (PDE) is an extension of the convection-diffusion-reaction (CDR) partial differential equation (PDE), which can be termed a mixed hyperbolic-parabolic PDE, with a nonlinear convection term and a first and third order source term. A numerical solution is computed by the method of lines (MOL), includin...

The Kolmogorov-Petrovskii-Piskunov equation, also called the Fisher-KPP equation or just the KPP equation, is a one-dimensional (1D) diffusion equation with a linear source term and a source term of arbitrary order (the order is a parameter). This chapter computes a numerical solution by the method of lines (MOL), including detailed discussion of t...

Partial differential equations (PDEs) are a general starting point for mathematical modeling and computer-based analysis throughout all of science, engineering and applied mathematics. Computer-based methods for the numerical and analytical solution of PDEs are therefore of broad interest. This chapter discusses some of the general approaches to th...

The Burgers-Huxley partial differential equation (PDE) is an extension of the convection-diffusion-reaction (CDR) partial differential equation (PDE), which can be termed a mixed hyperbolic-parabolic PDE, with a nonlinear convection term and a third and fifth order source term. A numerical solution is computed by the method of lines (MOL), includin...

The Kuramoto-Sivashinsky equation has a nonlinear convection term and second, third and fourth order spatial (boundary value) derivatives. Thus, this example illustrates the solution of a higher (fourth) order PDE. The four required boundary conditions (BCs) are taken from the analytical solution as two Dirichlet BCs and two Neumann BCs. A numerica...

This chapter combines a partial differential equation (PDE) with just a first-order (convective, hyperbolic) derivative in x and a PDE with just a second-order (diffusive, parabolic) derivative in x into a PDE that also includes a first-order reaction term. This convection-diffusion-reaction (CDR) PDE, which can be termed a mixed hyperbolic-parabol...

The regularized long-wave (RLW) equation has a linear and a nonlinear convection term, and a mixed partial derivative, first order in the initial value variable and second order in the spatial (boundary value) variable. A method of lines (MOL) analysis of the PDE leads to a system of coupled ODEs for which a numerical solution consists first of unc...

This chapter is the first pertaining to a PDE second order in the initial value variable, the hyperbolic Liouville equation, which has a second order derivative in the spatial (boundary value) variable and an exponential source term. A numerical solution is computed by the method of lines (MOL), including detailed discussion of the Matlab routines...

The Fitzhugh-Nagumo (F-N) partial differential equation (PDE) is an extension of the convection-diffusion-reaction (CDR) partial differential equation (PDE), which can be termed a mixed hyperbolic-parabolic PDE, with a linear and a cubic source term. The BCs include a single pulse and a train of pulses in time. A numerical solution is computed by t...

This chapter discusses the Boussinesq equation that has: (1) a second order derivative in the initial value variable, (2) second-order derivatives in the spatial (boundary value) variable with respect to the dependent variable and the square of the dependent variable, and (3) a fourth order derivative in the spatial variable. The four required boun...

The Kawahara equation has a nonlinear convection term, and a third and a fifth order spatial derivative. Thus, this example illustrates the solution of a higher (fifth) order PDE. Although technically five boundary conditions (BCs) are required, the spatial domain is specified to be long enough that the solution and its derivatives do not depart fr...

The mth-order Klein-Gordon equation has a second order derivative in the initial and spatial (boundary value) variables, a linear source term and an mth-order source term. When m is two, this is the quadratic Klein-Gordon equation; when m is three, it is the cubic Klein-Gordon equation. When both source terms are dropped, it is the linear wave equa...

The sine-Gordon equation is the classical wave equation with a nonlinear sine source term. This chapter computes a numerical solution by the method of lines (MOL), including detailed discussion of the Matlab routines and the numerical and graphical output. In this chapter, a series of mathematical transformations is applied to the sine-Gordon equat...

The term Lax Pairs refers to a set of two operators that, if they exist, indicate that a corresponding particular evolution equation is integrable. They represent a pair of �differential operators having a characteristic whereby they yield a nonlinear evolution equation when they commute. The idea was originally published by Peter Lax in a seminal...

A Bäcklund transformation transforms a nonlinear partial differential equation into another partial differential equation. Thus, a solution to the second partial differential equation must be compatible to the first partial differential equation. Hence, application of the Bäcklund transformation can provide a powerful method for generating solutio...

This paper provides an overview of the Hirota direct method which was first published in a paper by Hirota in 1971. The introduction of this approach provided a direct method for finding N-soliton solutions to non-linear evolutionary equations.

The one-dimensional (1D) diffusion equation, also termed Fourier's second law or Fick's second law is a basic parabolic partial differential equation (PDE) that admits traveling wave solutions. We first demonstrate how an assumed Lagrangian change of variable transforms the PDE to an ordinary differential equation (ODE) that can be integrated analy...

The partial differential equation (PDE) analysis of convective systems is particularly challenging since convective (hyperbolic) PDEs can propagate steep fronts and even discontinuities. To demonstrate this characteristic, we consider in this chapter the numerical and analytical integration of the linear advection equation, possibly the simplest PD...

The view has long been held by historians of science, that Sir Isaac Newton's original derivation of the inverse square law of gravity, whilst certainly not lacking brevity, most definitely provides little indication of the original thought processes that led him to the final results. For approximately three hundred years scholars have painstakingl...

A new release of an Introductory Global CO2 Model is now available. The model is intended to quantitatively introduce the CO2 problem at a basic level, with particular emphasis on ocean acidifica- tion which has not received the same attention as global warming and climate change, but could be just as important. The model is based on seven reservoi...

We report the difference in the solution of two tubular reactor PDE models in which the continuity (mass) balance:(1)includes only the reactant partial pressure, as reported in the literature, and(2)a revised continuity balance that includes reactant molar density (i.e., a nonlinear function of reactant partial pressure and temperature) throughout...

This paper describes a method for the state estimation of nonlinear systems described by a class of differential-algebraic equation models using the extended Kalman filter. The method involves the use of a time-varying linearisation of a semi-explicit index one differential-algebraic equation. The estimation technique consists of a simplified exten...

This paper describes a method for dynamic data reconciliation of
nonlinear systems that are simulated using the sequential modular
approach, and where individual modules are represented by a class of
differential algebraic equations. The estimation technique consists of a
bank of extended Kalman filters that are integrated with the modules.
The pap...

This paper describes a method for dynamic data reconciliation of nonlinear systems that are simulated using the sequential modular approach, and where individual modules are represented by a class of differential-algebraic equations. The estimation technique consists of a bank of extended Kalman filters that are integrated with the modules. The pap...

In this paper a method is presented for dynamic data reconciliation of nonlinear systems described by a class of differential-algebraic models, using the extended Kalman filter. A time-varying linearization is derived for a semi-explicit index one DAE. A simplified extended Kalman filter algorithm is then presented, and its integration with the DAE...

In this paper the implementation of dynamic data reconciliation
techniques for sequential modular models is described. The paper is
organised as follows. First, an introduction to dynamic data
reconciliation is given. Then, the online use of rigorous process models
is introduced. The sequential modular approach to dynamic simulation is
briefly disc...

In industrial practice, constrained steady state optimisation and predictive control are separate, albeit closely related functions within the control hierarchy. This paper presents a method which integrates predictive control with on-line optimisation with economic objectives. A receding horizon optimal control problem is formulated using linear s...

In industrial practice, constrained steady state optimisation and predictive control are separate, albeit closely related functions within the control hierarchy. This paper presents a method which integrates predictive control with on-line optimisation with economic objectives. A receding horizon optimal control problem is formulated using linear s...

A predictive control algorithm which attempts to optimise the steady-state operating conditions of the process is presented. The algorithm is based on an adaptive linear state space model of the process together with receding horizon optimal control solutions. The algorithm is applied for the optimisation of a simulated chemical reaction system.

An optimising controller that is able to drive a plant from a suboptimal operational condition to its steady-state optimum based on receding horizon optimal long range predictions is presented. The technique has been tested wilh realistic simulations of an industrial distillation column using a rigorous process simulator.

A real-time dynamic two-phase model of the pipeline system and relevant portions of shore facilities is being installed for Woodside Offshore Petroleum Pty. Ltd.'s North West Shelf gas project. The trunkline management system (TMS) which utilizes dynamic pipeline and plant process simulation is scheduled to start up near the end of the first quarte...

A trunking management system (TMS) utilizing dynamic pipeline and plant process simulation is scheduled to start up near the end of first quarter 1994 at Woodside Offshore Petroleum Pty. Ltd.'s North West Shelf gas project at Karratha, Western Australia. The TMS is a real-time, dynamic two-phase model of the pipeline system and relevant portions of...

The need for improving the online adjustment of process operating conditions has long been recognised as an important issue of plant operation and management in the chemical process industry. In order to address this need, substantial research into improving existing optimisation algorithms has been performed during the last decade; this has lead t...

Historically, process designers using the ICI process have used a fixed bed quench type reactor based on a mid-1960s design. Davy McKee uses a slightly modified version of this quench ''converter''. The methanol synthesis reaction liberates heat and in order to moderate temperatures, cold gas is injected into the catalyst bed at various levels. In...

Prior to the Middle East war, 'state of the art' low pressure methanol plants operated with an energy consumption of about 34 million BTU plus 55 kWh per ton of methanol product. Today's plants consume less than 27 million BTU per ton whilst at the same time providing all their own electrical power requirements. This paper describes the sequence of...

A procedure is given for designing multivariab1e control systems in the direct Nyquist plane using the principle of diagonal dominance. It is shown that Bode technlques can be extended to the mu1tivariab1e case and that dominance can be verified by using straight line approximations to the frequency response. In addition, Rosenbrock's idea of using...

This paper describes the instrumentation and control of methanol plants and relates to past experience and future developments. Brief descriptions of the reforming and methanol synthesis processes are given. Past experience relates to large throughput plants, designed for severe ambient conditions and remote locations.