Potential Honours Projects

The following list does not aim to be an exhaustive list of honours projects that I am interested in supervising, it is a list of things that are interesting to investigate. Don't worry if some of these projects don't make to you, come and see me for more details - the comments make sense to me.

• Autothermal reactors.
• Biological Heating.
• Biotechnology.
  • Activated Sludge Model
  • Andrews Model
  • Contois Model
  • Double substrate kinetics
  • Ethanol and biodiesel
  • Enzyme Reactions
  • Immobilised Enzyme Reactors
  • Large models
  • Membrane Reactors
  • Methane prodution in an anaerobic digestion system.
  • Incomplete mixing
  • Nano-particulate Membrane Bioreactor (NMB)
  • Predator-prey dynamics
  • Processing of industrial wastewaters and slurries
  • Simple models for product inhibition
  • Seperator-reactor systems using a membrane
  • Sludge Disintegration
  • Standard Model
  • Substrate Inhibition
  • Miscellaneous
• Combustion
• Drug delivery by nano-particles
• Food Engineering.
• Thermal sterilisation of Canned Food
• Drying
• Freezing of Ice-Cream
• Protecting probiotic bacteria in the intestines
• Heat Transfer.
• Heterogeneous catalysis
• Mathematical Ecology and Epidemiology
• Mathematical modelling of the GIT
• Multiple Scales
• Non-isothermal reactors
• Semi-analytical solutions
• Transform methods to solve linear PDEs

Autothermal reactors

Autothermal processes combine exothermic reactions, which produce heat, and endothermic reactions, which take heat out of the system. The advantages of autothermal reactors include:

1. In conventional reactors energy must be supplied for endothermic reactions to proceed, typically through the use of external burners or interstage heaters. The exothermic reaction produces heat, increasing the temperature of the reactor and increasing the rate at which the endothermal process occurs. In an autothermal reactor we get the heating for free. The absence of an external burner and the accompanying power supplies makes the system both simpler and less expensive.
2. The endothermal reaction controls the temperature of the system, making it less likely that thermal runaway will occur as it provides a source of cooling.
3. If the system is correctly setup then the overall process is energy neutral. The energy released by the exothermic reaction drives the endothermic reaction. This is an autothermal reactor.

There is growing interest in the use of autothermal reactors to efficiently produce hydrogen from natural gas, and other fuels such as ethanol, for use in fuel cells. Such technology has great potential as an energy supply for compact mobile power supplies, which can be used in homes and cars of the future. It can also be scaled up to provide a distributed supply of energy for industrial markets. The use of such reactors will contribute towards reducing greenhouse gas emissions.

Hydrogen can be produced through the catalytic reforming of hydrocarbons. This process couples endothermic steam reforming with exothermic oxidation to create hydrogen-rich fuel feeds. In this project we will examine some simple models for this process.

Many chemical engineering processes can be modelled as endothermic and exothermic reactions either in parallel, or in series or in competition. Such reaction schemes, modeled by relatively simple systems of non-linear differential equations, give raise to complicated behaviour.

Biological Heating

Generation of heat within the human body. References supplied by Professor Brian Gray are:
Biological Applications of Combustion Theory, with N.A. Kirwan and P. Gray. Combustion and Flame 18, 439, 1972.
Heat Generation in Tissues, Bulletin of Mathematical Biology, 42, 273 (1980).
Distribution of Heat Sources in the Human Head, Journal of Theoretical Biology, 82, ???
Surface law and Metabolic Rate. J. Theoretical Biology, 2, 757, 1982.
Physical Theory of Enzyme Catalysis (with I. Gonda). J. Theoretical biology, 94, 513 (1982).

The last one is quantum mechanics, so you might not want to know about that.

Wet combustion. Build proper models for moisture transport, evaporation, condensation into the model. Biology+chemistry. Reaction-Engineering Model.

Biotechnology

• Panikov, N.S. Microbial Growth Kinetics. Chapman & Hall, UK, 1995.
• Identification of ODE Models.
  

  Ordinary differential equation (ODE) models have been widely used to model physical phenomena, engineering systems, economic behaviour, and biomedical processes. In particular, ODE models have recently plyaed a prominent role in describing both the within host dynamics and epidemics of infectious diseases and other complex biochemical processes. Great attention has been paid to the so-called forward problem or simulation problem, i.e.\ predicting and simulating the results of measurements or output variables for a given sustem with given parameters. However, less effort has been devoted to the inverse problem, i.e., using the measurements of some state or output variables to estimate the parameters that characterises the system, especially for nonlinear ODE models without closed form solutions.
  H. Miao et al. SIAM Review 53(1), 3-39, 2011. 3-39, 2011.

  Suppose that we have a mathematical that contains n ODEs. We can measure m (m≤n) output variables. The model contains p parameters. Can we estiamte the p parameters from measurements on the m output variables?

  Idea has been applied to Monod model (page 10). There are plenty of other models to try.

Activated Sludge Model

• A system with two tanks, one aerobic, one anaerobic and one settler.

Andrews model

1. Look at double-substrate inhibition models with constant yields. (Follow up to Tim's advanced maths project with emphasis on stability calculations). (wu-sc:2007)
2. Variable yield model with death. suzuki:1985.
3. grieves:1968 Look at two-tank model with variable volume distribution and variable feed distribution.

Contois model

• Hydrolysis of substrate (Contois) followed by Monod reaction. Could be two bacteria or the same. See papers by Gawande, Ramirez.
• Two microbial species competing for one substrate.
• What if the substrate follows a logistic equation so this is not a chemostat?

Double substrate kinetics

(a good keyword). See sonmezisik:1998. "Under certain conditions, the growth rate of an organism may be simultaneously limited by two or more substrates" (6-9).

beyenal:2003 Double substrate kinetics that are both Tessier. (Standard is to use monod for both).

yurt:2002 "It is well-known that the microbial growth rate often depends on more than one substrate (Machado et al 1989; Beyenal and Tanyolac, 1997; Neeleman et al, 2001).

Ethanol and biodiesel

A fancy phrase of biodiesel production is biotechnology oil production.

• Examine models for the production of ethanol through the growth of Saccharomyces cerevisiae growing on ethanol. (jones-kd:1999)
• tyagi:1980 has a model with substrate and product inhibition with a four reactor cascade.
• ghose:1979a recyle experiments (interesting to assume that all product is extracted from the recycle process).
• ghose:1979b substrate limitation and product inhibition.
• dourado:1987 distributed feeding is a way of removing the negative effect due to substrate inhibition... In a cascade reactor, part of the fermentation is carrief out in a low-ethanol-concentration environment (the first reactors)... A cascade of reacors allows us to distribute the global dilution rate and the input substrate among several reactors. First, the substrate concentration in each reactor can be adjusted in order to reduce substrate inhibition... The advantage of a cascade reactor is that it allows a substantial increase in ethanol productivity... The distributed feeding appears to be interesting for two main reasons. First, it allows us to eliminate the drawback of the washout state if one is interested in a low output ethanol concentration. Second, if the output ethanol concentration is high, distributed feeding is profitable when the model is strongly substrate inhibitory.
• Substrate inhibition continued. Repeat Hill and Robertson (1989) but consider multi-stream feed strategies.
• economou:2010: batch model for biodiesal production.

Enzyme reactions

• M. Stamoszewski and S. Koter. Theoretical analysis of steady state for ester hydrolysis in an enzymatic membrane reactor with production retention. Desalination, 248 (2009) 224-234.
• staniszewski:2009
  ester hydrolysis when the maximum growth rate depends upon the pH of the solution which in turn depends upon the product concentration.
• R. Suresh and M. Chidambaram. Periodic operation of well mixed enzyme reactors with steady state multiplicites using relay feedback. Bioprodess Eng, 16 (1997), 225-227.
• P.-Y. Ho and H.-Y. Li. Determination of multiple steady states in an enzyme kinetics involving two substrates in a CSTR. Bioprocess Eng., 22 (2000) 557-561.
• J. Vasquez-Bahena, M.C. Montes-Horcasitas, J. Orega-Lopez, I. Magana-Plaza and L.B. Flores-Cortera. Multiple steady-states in a continuous stirred tank reactor: an experimental case study for hydrolysis of sucrose by invertase. Process Biochem., 39 (2004) 2179-2182.

Immobilised Enzyme Reactors

justification purwadi:2008

Large models

• ADM1 model. See ramirez:2010.

Membrane Reactors

Incomplete mixing

Methane prodution in an anaerobic digestion system

A simple model for the dynamics of an anaerobic wastewater treatment system is to treat it as two reactions: acidogenesis and methanogenesis.

aS -> cA +X
dA -> CH4 +M
S is the organic substeate, A is the volatile fatty acids, X is acidogenesis bacteria, CH4 is methane and M is methanogenesis bacteria.

In the first reaction, the acidogenic bacteria (X) consume the organic substrate (S) and produce volatile fatty acids (A). In the second reaction, the methanogenic bacteria (M) use the volatile fatty acids as substrate for growth and produce methane. The anaerobic digestion process must be operated such that the acidification (accumulation of volatile fatty acids) of the reactor is avoided.

Incomplete mixing

Apply ideas to blood coagulation model. pompano:2008.

Nano-particulate Membrane Bioreactor (NMB)

AMT have developed a nano-particulate membrane bioreactor (NMB) which has been shown to work at various scales with gray water, grease trap waste, sewage and many different industrial waste water streams such as effluents from breweries, wineries and a detergent factory. We will develop models from this process based upon partial differential equations. (See work folder for chemeca flyer). (Note basic membrane reactor model also appears in yoon-sc:2004).

Predator-prey dynamics

Food processing wastewaters and slurries typically contain high concentrations of biodegradable organic matter. Before the wastewater can be discharged the pollutant concentration must be reduced. One way to achieve this is through the use of a biological species (`biomass') that consumes the organic matter (`substrate').

In this project we will examine what happens when competition is moved up one level by investigating what happens if there is a predator that grows by consuming the biomass. Our interest is in how this competition influences the level of pollutant that is discharged from the reactor.

• diffusion models. "Predator inhibition models". hsu-su:1978 papers.
• What happens if there are two species competing for one food source (Monod model, the `weaker' one must become extinct) but the weaker species is also a predator?

• A biological wastewater treatment process can be considered as an artificial ecosystem, and activated sludge is an ideal habitat for several organisms other than bacteria. One way to reduce sludge production is to exploit higher organisms such as protozoa and metazoa in the activated sludge processes that predate on the bacteria whilst decomposition of substrate remains unaffected... In an activated sludge system the grazing fauna mainly consists of protozoa and occasionally metazoa. Protoa present <1% of the total dry weight of a wastewater biomass, and 70\% of protozoa are ciliates [6,88]. The protozoa present can be divided into four groups: ciliates (free swimming, crawling and sessile),flagellates, amoeba, and heliozoa [89].

  It is well known that the presence of protozoa and metazoa in aerobic wastewater treatment processes plays an important role in keeping the effluent clear by consuming dispersed bacteria... Recently, many researchers have focused on sludge reduction induced by grazing on bacteria [90-107]. wei-y:2003

• The Two-stage system considered in wei-y:2003 with predators only in the second stage.
• Periodic forcing.
• COnsider the classic three-step food chain but allow the substrate to be another animal species with a decay rate. How does this effect the behaviour of the system?
• In a membrane reactor. Perhaps the membrane reactor only applies to one of the species or perhaps it applies to both species?
• Predator-prey with variable yield.
• X -> substrate + particulates and predators!

Another application is to reduce excess sludge production in the activated sludge process - zhang-b:1996

Sludge disintegration

1. Sludge disintegration inside a normal bioreactor. EB

   Yasui and Shibata (wei-y:2003, 31) developed a new process for reducing sludge production in the activated sludge process. The process consists of a sludge ozonation stage and a biodegradation stage, in which a fraction of recycled sludge passes through the ozonation unit and then the treated sludge is decomposed in the subsequent biological treatment.

2. Add a model for sludge disintegration. Thomas' model should be extended from 3 ODE to 6 ODE. The sludge disintegration unit should be treated as a second reactor. EDB
3. Easy project is to look at exact parametric sensitivity in the yoon model using the exact steady-state values.
4. EDB. Note that the usual model for sludge disintegration assumes that μ_m and K_S are the same for all soluble organic materials, whether they are originated from influent or disintegrated liquor.
5. The ultimate sludge-disintegration model is to use the bioreactor model from yoon-sh:2005 with simple kinetics.
6. Should look at a disintegration unit inside a normal bioreactor.
Variation of MLSS in a cascade of reactors with sludge disintegration units?
7. grady:1980chap12 might be useful for background models with nonviable cells.

Processing of industrial wastewaters and slurries

1. Re-consider the model with a variable yield coefficient and non-dimensionalise it correctly. Add death to the model.
   1. Determine performance of the optimised cascade and the cascade with equal residence time for two and three reactors.
   2. Re-investigate membrane model. H21 bifurcation curve in alpha-beta-k_d plane. (note basic membrane bioreactor model also appears in yoon-sc:2004)
2. 1. Model of a fixed-bed biological process and a generalized model of a fixed-bed biological process
   2. The fixed-bed reactor as two completely mixed compartments in parallel (Escudie et al., 2005).
3. Contois model with X₀ not equal to zero. EB. stress related death in a model with oxygen transfer (Rubayyi). replace V*k*X by V*(k+a*KLA)*X where KLA is the oxygen mass transfer coefficient. Idea is that the intense mixing needed for oxygen transfer increases the stress on the biomass. see cliffe:1988 (page 280)
4. harmand:2006b looked at the standard model with recirculation and by-pass in a control session. Can this be extended to the contois model?
5. Investigate a model with a sludge disintegration system (cf thomas' masters project)
6. Ice cream in a series of membran reactors.
7. Use Andrews solutions for a two- and three- reactor cascade. Use the optimisation functions of Erickson and Fan (1968). This is really an optimisation project, rather than a dynamical systems project. Similarly can look at the paper by Grieves and Kao. And the paper by Scuras (2001). And Hill and Robertson (1989). Harmand et al (2003) is definitive for some systems. Could re-investigate the effect of death upon many of these systems. It's interesting the in the paper with Andrew I observed that at high residence times equal-residence times give a performance that is very nearly optimal. Optimising the reactor design only seems to bring improvements at low residence time. Is this a function of death? Investigate recycle. Good
8. grady:book page 640. Step Aeration Activated Sludge (SAAS) can be modeled as four CSTR's in series with feed distribution to each tank

Simple models for product inhibition

• Production of ethanol in multiple reactors including cell-recycle. Use a simple product inhibition model. How do we estimate parameter values from experimental data?
• Add recycle to current model.
• Optimisation of reactors, see Hill and Robertson (1989). Could compare different product inhibition mechanisms.
• wall:1992 A good approximation to this phenomenon is the linear inhibition model of Ghose and Tyagi (1979).
  μ = μ_mS/(K_S+S)*(1-P/K_I)
  where K_I now represents the maximum ethanol concentration above which growth ceases. Optimisation in a sequence of tanks?
• zhang-j:2009. Death Ethanol is formed to not only inhibit the specific growth rate, but also to acclerate death.
  k = k_d +alpha*exp(D*[ethanol]).
• Hill function. See ramirez:2010.
• ishizaki:1995. The kinetic parameters for substrate/microorganism/product (eg \mu_max, K_s, K_i) could be different.
• Re-examine models including direct product formation from the substrate.
• models with non-growth associated product formation dinopoulou:1988

Seperator-reactor systems using a membrane

Extend Soji model to consider different kinetics: Monod, Contois, Andrews and product inhibition. (note basic membrane bioreactor model also appears in yoon-sc:2004)

Standard Model

1. Add a model for hydrolysis of death biomass. EDB
2. Oxygen as a variable with oxygen transport. stress related death in a model with oxygen transfer (Rubayyi). replace V*k*X by V*(k+a*KLA)*X where KLA is the oxygen mass transfer coefficient. Idea is that the intense mixing needed for oxygen transfer increases the stress on the biomass. see cliffe:1988 (page 280)
3. Model wall growth.
   1. bungay:1968 "In our laboratories, some studies of competition have shown that adherence to the walls of the vessel can be a key factor in competition. A slower growing species can persist in appreciable numbers in competition with rapidly growing organisms if the slow grower is continually reinoculated from the wall growth into the main liquid bulk".
   2. senn:1994 "It is well-known that wall growth can affect steady-state concentrations in chemostat cultures. This effect should be most pronounced at low input substrate concentrations in the medium feed where the proportion of biomass on the wall is high compared to that in solution".
4. canale:1973 "a massive protozoan lysis had occurred. This cell lysis caused considerable amounts of debris to accumulate within the medium".
   Two other continuous runs of short duration at low dilution rates resulted in a similar breakdown of the Tetrahymena population.
   See predator-prey notes for a simple model.
5. Models in which both species produce a toxin for each other with a variable yield coefficient.
6. abulesz:1987 investigated time delay models. This leads to a system of three, rather than two, equations. This opens up a large avenue of potential problems!

   "All unstructured models predict a response to step-changes in operating variables which is faster than experimentally observed (40). This is a result of the inherent assumption of those models that there is no time lag between changes in the substrate level and adjustment of the growth rate at the appropriate level. To relax this assumption, one may assume that the specific growth rate is a function not only of the present substrate level but also of previous levels in a weighted manner". See also their references (41 & 42) and page 1062.

7. 1. beyenal:2003 "It was expected that at low agitation rates the growth of microorganisms was limited by external mass transport. Therefore, when the agitation rates increased, the mass transfer rate to the microorganisms increased, along with the SOUR, which reached a maximum value. Increasing the agitation rate beyond this maximum actually decreased the SOUR, probably because the agitation was injuring the microorganisms."
   2. yurt:2002 At low agitation rates, Leptothrix discophora SP-6 aggregated, and at higher agitation rates it was mechanically damaged.
8. Is it possible to investigate a stochastic version of the standard model and investigate the effect on wash-out conditions?
9. Standard model with dead cells. See grady:1980 for background and also link this to the sludge disintegration unit work.
10. grady:1980. Basic CSTR model with non-viable cells.
11. grady:1980. Recycle models. There are four cases of particular interest, all of which will use equal reactor volumes. In the first case, the system corresponds to a simple chain, with all feed and all recycle going to reactor one. In case two, the feed and recycle are distributed evenly amongst the four tanks. In case three, the feed is distributed evenly among the four tanks and the recycle is added to tank 1, whereas in case four, all recycle is returned to tank 1 and all feed enters tanks 3.
12. grady:1980. Extend the basic model to analyse oxygen demand. This can then be used to determine waste stabilisation. (chapter 12). This would make a good six credit-point project.
13. Velocity dependent death rate. Speak to John Kavanagh about "shear intolerance".
14. Asymptotics for membrane reactor cascade (contois). Compare against conventional reactors.
15. jost-c:2000 has lots of interesting things to read.

    The first approach to reconcile theory and experiment was to introduce flexible models that contain both Monod's and Contois's functions as special cases (Roques et al 1982, Borija et al. 1995).

    μ(S,X) = μS/(K_s+S+cX)

    This form was introduced independently in ecology by DeAngelis et al (1975) and by Beddington (1975).

16. incomplete mixing using the standard approach. First assume no death.
17. Incomplete mixing. Allow epsilon to be a function of delta. epstein:1995
18. roques:1982 proposed a general rate expression which includes both the Monod and Contois expressions as special cases:
    μ(S,X) = &mu_maxS/(S+a+bX).
19. Model recycle more realistically (if possible) by looking at sludge thickness analysis. See references [62,65] on page 673 of grady:book for starts. SIAM paper. two substrates and two micro-organisms. How does long-term behaviour depend upon parameter values?
20. bush:1976 death-rate ``is assumed to be independent of the substarte concentration''.
21. substrate-dependent death. huang-d:2011 is a cracker.

Substrate Inhibition

Miscellaneous

Food Engineering

Thermal sterilisation of Canned Food

This project will investigate models for the sterilisation of canned food. It will involving solving PDEs numerically. See mohamed:2003 for references.

Drying Food

Dong suggests the following as a good research problem for a strong PhD student

My suggestion is to work in heat/mass transfer (in drying) coupled with mechanics (stress strain analysis)...this can make a difference in literature. The ideal candidate is to look at how micron sized particles (functional particles for medical or food purpose) formed during liquid removal. So far heat and mass transfer have been investigated a great deal but the formation of the shell and shell structure has not been touched which I feel very very important.

Freezing of Ice-Cream

Ice-cream is quite an interesting substance (as well as one that is nice to eat!). This project will investigate PDE models for the manufacture of ice-cream.

Protecting probiotic bacteria into the gut

For orally administered live bacteria (probiotics) to function, they must be protected from the high concentration of bile acids that are found in the intestine. This is essential to ensure reproducible and efficient live cell delivery.

This project will use reaction-diffusion models to investigate how the incorporation of bile adsorbing resins into capsulated drug delivery systems protects probiotic bacteria.

Heat Transfer

• This is a numerical PDE's project. You will model heat-transfer through the human skin in order to estimate how long an individual can be exposed to a specified heat-flux before the skins develops burns of a specified intensity. (The model already exists. The aim of this project is to code up the model).

Heterogeneous catalysis

Extend CTM paper on heterogeneous catalysis from a single reaction to a bi-molecular reaction.

Mathematical Ecology and Epidemiology

1. 2 predator-1 prey model (hsu-sb:1978 papers) with diffrent forms for the diffusion terms. Perhaps even the 1 predator model is of interest?
2. R.S. Cantrell and C. Cosner. Spatial Ecology via Reaction-Diffusion Equations. Wiley. 577.0151/3.
   Malchow, S.V. Petrovskii, Venturino. Spatiotemporal Patterns in Ecology and Epidemiology. 577.015118/2
   S. Cantrell, C. Cosner and S. RUan. Spatial Ecology. 577.015118/5.
3. Apply semi-analytical technique to RD equations.
4. jost-c:2000 has lots to say about predator dependent growth-rates in mathematical ecology in analogy with microbiology and provides good references.
5. Brindley-Truscott model. Contois formulation? PDE formulation applying semi-analytical technique?
6. tian:2011. Turing patterns created by cross-diffusion for a {Holling II} and {Leslie}-{Gower} type three species food chain model
7. Predator-prey models. Predators might diffuse up the prey gradient.
8. Competition for nutrients and other resources is an interaction common among microbial species growing together in the same environment. Competition tends to eliminate species from the system. The main question then is whether the competing microbial species can coexit and under what conditions. pavlou:2013

Combustion

Follow-up on Dong's heat-test method. Look for solutions to the linear problem? HAM solutions?

Drug delivery by nano-particles

1. Spherical particles with concentric layers of inert and drug. Can target time-to-release by design of layers.
2. Maybe c(r) - internal concentration of drug as a function of radius as a way to control drug delivery rate.
3. Want to achieve a constant delivery of drug.
4. Maybe we could design different biodegradable materials with different k values. Then distribution of k values leads to a different delivery profile.

Food Engineering

The food industry offers a surprisingly rich variety of interesting problems in applied mathematics. Here are a couple that I am interested in.

Drying of small particles

We will investigate the composition and temperature of spray-dried particles using distributed drying kinetics. This model will be used to investigate segregation, which has been observed experimentally during the spray drying of milk. (Model can also be applied to the rice swelling problem.)

Drying of a French Fry

This project involves modelling the frying of a french fry in hot oils. The primary practical interest in this problem is to drive vapour out of the fry and this is what we will seek to model. The chicken pattie model may be useful for this problem...

Mathematical modelling of the GIT

1. Volume increasing + decreasing. Fit data to wrong models? EB?
2. Spratt, P., Nicolella, C., Pyle, D.L. (2005) An engineering model of the human colon. Trans IChemE, Part C, Food and Bioproducts Processing 83 (C2), 147-157.
3. Develop a model for how the pH changes in the human stomach following a meal. Combine this with a model for the delivery of a particulate drug. How do certain gastro-disease effect the delivery of the drug?
4. Investigate how the diffusion of bile (secreted into the small intestine) into a tablet kills dried bacteria. Investigat mechanisms for retarding the uptake of the bile by the bacteria. You will need to learn numerical methods for solving PDEs!

Dong: I think if u can incorporate the detailed chemistry in the stomach and also some dimensional effect such as the breakdown of the encapsulated materials...that shoulod be meaty enough.

Multiple Scales

Contract Troy!

Non-isothermal reactors

Many chemical engineering processes can be modelled as endothermic and exothermic reactions either in parallel, or in series or in competition. Such reaction schemes, modeled by relatively simple systems of non-linear differential equations, give raise to complicated behaviour.

Classic first-order non-isothermal reactor PLUS non-perfect mixing models.

Semi-analytical solutions

This would be a joint-project with Professor T.R. Marchant.

• Problems with prescribed flow conditions. John Dold's talk at the John Clark meeting.
• Biochemical Engineering Book. Chapter 7.3.
• Gray-Scott model. Extensions from citations to papers by Lin.
• reactive flows. Dold's papers. Look for "vortices-flow".
• blood coagulation model from pompano:2008
• Aris books
• S.P. Hastings. Classical Methods in Ordinary Differential Equations. American Mathematical Society. (maybe, need to look at it).

See also Mathematical Ecology and Epidemiology

Transform methods to solve linear PDEs

Dong's salt diffusion problem as a two-layer diffusion problem. Apply the mean action time approach used by Hickson.