analytical solution of population balance equation ... solution of population balance equation...

Appl. Math. Inf. Sci.9, No. 5, 2467-2475 (2015) 2467

Applied Mathematics & Information SciencesAn International Journal

http://dx.doi.org/10.12785/amis/090530

Analytical Solution of Population Balance EquationInvolving Growth, Nucleation and Aggregation in Termsof Auxiliary Equation Method

Zehra Pınar1, Abhishek Dutta2,3, Guido Beny2 and TurgutOzis4,∗

1 Department of Mathematics, Faculty of Arts and Science, Namık Kemal University, 59030 Merkez-Tekirdag, Turkey.2 Faculteit Industriele Ingenieurswetenschappen, KU Leuven, Campus Groep T Leuven, Andreas Vesaliusstraat 13, B-3000 Leuven,

Belgium.3 Departement Materiaalkunde, KU Leuven, Kasteelpark Arenberg 44 bus 2450, B-3001 Heverlee-Leuven, Belgium.4 Department of Mathematics, Faculty of Science, Ege University, 35100 Bornova-Izmir, Turkey.

Received: 31 Jan. 2015, Revised: 2 May 2015, Accepted: 3 May 2015Published online: 1 Sep. 2015

Abstract: The Auxiliary Equation Method (AEM) has been modified to obtain the solutions of a Population Balance Equation (PBE)involving particulate growth, nucleation and aggregationphenomena. In all the cases examined, the volume density distributions areaccurately predicted by the travelling wave solutions of the complementary equation of the nonlinear partial integro-differential equationwith distinctly chosen parameters. Being a flexible technique and a direct comparison for the existing analytical solutions, this studyproves the potential of the proposed methodology.

Keywords: Population Balance, Growth, Nucleation, Aggregation, Auxiliary Equation method, Travelling wave.

1 Introduction

Particulate processes are characterized by sizedistributions that are assumed to vary strongly in timewith respect to mean particle size and shape of theParticle Size Distribution (PSD). Population balances arewidely encountered in numerous scientific andengineering disciplines to describe the evolution of PSDin processes that involve particulate phenomena likegrowth, nucleation, aggregation and breakage [1]. Theseparticulate phenomena may occur during the processwhich may result in momentous changes manifestedthrough its PSD. The time evolution of this distribution isdetermined by the solution of the so-called PopulationBalance Equation (PBE) which governs the dynamicbehavior of particulate processes through a nonlinearpartial integro-differential equation within themathematical framework [1]. The numerical solution of adynamic PBE is a remarkably complicated problem dueto both numerical complications and the uncertainties ofthe model regarding the particulate mechanisms that arefrequently weakly approximated. On the other hand, the

essence of any numerical solutions of a PBE requires thediscretization of the particle diameter/volume domain bymeans of certain numerical approximations that results ina system of stiff, nonlinear integro-differential equations.

Since the early 1960s, various numerical methodshave been developed for solving PBM involving bothtime-dependent and -independent formulations. Amongthose include the fully discrete method [2,3], method ofclasses [4,5], fixed and moving pivot method [6,7],higher order discretized method [8,9], orthogonalcollocation on finite elements [10,11], Galerkin andwavelet-Galerkin method [12,13], Monte Carlo [14] andleast squares method [15,16]. Several other techniquesavailable for solving the PBEs are the various method ofmoments where the PBE is solved for the moments of thePSD, e.g., quadrature method of moments (QMOM) [17,18] and direct quadrature method of moments (DQMOM)[19,20].

Despite numerous papers published on the numericalsolutions of a PBE, the choice of the most appropriatemethod for the calculation of time evaluation of a PSD, inparticular processes undergoing simultaneous particle

∗ Corresponding author e-mail:[email protected]

c© 2015 NSPNatural Sciences Publishing Cor.

http://dx.doi.org/10.12785/amis/090530

2468 Z. Pınar et al. : Analytical Solution of Population Balance Equationn...

growth, nucleation, breakage and aggregation, is notstraightforward and realistic. In reality, a large number ofstudies refer to only a limited range of variation ofparticle breakage and aggregation rates. Therefore, thewide-ranging application of a numerical method to thesolution of a specific problem cannot be assured. On theother hand, the formulation of a large number of differentnumerical approaches also underlines the intrinsicproblems in obtaining a precise and consistent numericalmethod. In this study, a one-dimensional (1D) populationbalance equation (PBE) under various conditions ofparticulate nucleation, growth and aggregation areinvestigated and compared with their analytical andnumerical solutions as found in the literature. Then theanalytical method introduced by Pınar andOzis [21,22] isimplemented to solve the PBE and the validity of thesolution is established through example case studiesinvolving growth, nucleation and aggregation processes.

2 Problem Formulation

Population balance modeling has become an importanttool to model a wide variety of particulate processes andits increasing applicability in various disciplines ofscience and engineering. For example, processes likecrystallization, granulation, milling, polymerization,flocculation, and aerosols involve population balances.The PBE describes the evolution of a density function,representing the behavior of a population of a state vectorsuch as size or volume of solid particles, liquid dropletsor gas bubbles. The rapid development in this field ispossible thanks to the availability of improved particlesize measurement techniques to measure multivariatedistributions. The evolution of this density function takesinto account the different processes that control thepopulation such as aggregation, breakage, growth andadvective transport of the state vector. The generalizedone-dimensional Population Balance Equation (PBE) in awell-mixed control volume, without considering spatialdependence, is written as:

∂∂ t

n(x, t)+∂∂x

[G(x, t)n(x, t)] = Anuc+Aagg+Abreak (1)

where

Anuc= B0(t)δ (x0)

Aagg=

∫ x

0n(x− x′, t)n(x′, t)a(x− x′,x′)dx′

−∫ ∞

0n(x, t)n(x′, t)a(x,x′)dx′ (2)

Abreak=

∫ ∞

xb(x,x′)n(x′, t)Γ (x′)dx′−Γ (x)n(x, t)

where then(x, t) is the number density function interms of the particle volumex. a(x,x′) is the

volume-based aggregation kernel that describes thefrequency at which particles with volumex andx′ collideto form a particle of volumex + x′, Γ (x) is thevolume-based breakage function and the stoichiometrickernel b(x + x′), satisfying the symmetry andnormalization conditions, gives the product sizedistribution for binary breakage through the probability offormation of particles with volumex from the breakage ofparticles of volumex′.

3 The Auxiliary Equation Method (AEM)approach

The Auxiliary Equation method (AEM) suggestedrecently by Pınar andOzis [21,22] has been applied tononlinear physical models and has successfully helped todevelop new analytical solutions with appropriateparameters and these parameters are chosen such a waythat the obtained solutions simulates the behavior of thesolutions of aforementioned problems. Recently, Pınar etal [23] proposed analytical solutions of PBEs involvingparticulate aggregation and breakage using AEM andcompared the results with their available analyticalsolutions obtained from the literature for a 1D PBE. Inthis study, the proposed methodology is further extendedto include growth and nucleation within the mathematicalframework. The detailed features of this method can beread from the recommended papers of Pınar andOzis [21,22]. For the sake of brevity, the adjustment of theanalytical method for the proposed solution is brieflymentioned here and the detail is left for the interestedreaders to refer to Pınar andOzis [21,22] and Pınar et al[23].

In the previous section, a particulate populationbalance is introduced briefly. In this study we shalldevelop an analytical particle distribution conjecturewhich works well in the case of certain well-definedstandard processes of particle formation. The distributionfunction is defined by Eq. (1) and the physical parametersaffecting the formation of the distribution are nested inthis differential equation. In our conjecture, we adapt thepreviously mentioned works of Pınar andOzis [21,22] todetermine the solution of the batch PBE problem. Ourconjecture is based on the reinterpretation of particlephase space and amalgamates it with the methodology assuggested in Pınar andOzis [21,22]. Particle phase spaceconsists of least number of independent coordinatesattached to a particular distribution that allow a completedescription of the properties of the distribution. Particlephase space and may conveniently be divided into twosub regions given by internal and external particlecoordinates. External coordinates refer simply to thespatial distribution of the particles. Such externalcoordinates of course are not necessary, for example, inthe description of a well-mixed particulate process,although it may be quite convenient to report the


Appl. Math. Inf. Sci.9, No. 5, 2467-2475 (2015) /www.naturalspublishing.com/Journals.asp 2469

distribution on a unit value basis. Internal coordinate referto those properties that are attached to each particle thatquantitatively measure its state, independent of itsposition. A common example of an internal coordinateproperty is particle volume. The particle volumedistribution is defined so thatn(x, t)dx is the number ofparticles per vessel volume at timet in the volume range(x,x+dx) and therefore the total volume=

∫ ∞0 n(x, t)xdx

must also be conserved [24]. In any particulate processgiving rise to the formation of PSD, individual particlesare continuously changing their position in the particlephase space, namely, each particle moves along thevarious internal and external coordinate axes. If thesechanges are gradual and continuous, one refers to thismovement as convection along the respective particlecoordinates and refers to the rate of change of thecoordinate property of a particle as the convective particlevelocity along that coordinate axes [24].

Hence, in this study, the particle velocity isconjectured and assume to be slow enough andcontinuous and the particle velocity components arenested in the functionζ = ζ (x, t) where x the particlevolume and timet. Because, internal convection velocitycan be taken as linear rate of growth of a particle andζ = ζ (x, t) can be chosen a linear combination of particlevolume x and the timet. An alternative to tracking theconvective particle velocity is to fix it by a suitable choiceof new space coordinates i.e.ζ = ζ (x, t) and solve theproblem in this new coordinate system. As such,following the methodology in Pınar andOzis [21,22] thenonlinear partial integro-differential equation i.e. thebatch PBE (Eq.1) can be readily reduced to ordinaryintegro-differential equation using an independent singlevariable ζ where ζ = ζ (x, t) is a function of particlevolume x and time t. Consequently Eq. (1) and thecomplementary equation are invariant under theappropriate transformationζ = ζ (x, t) and also theirsolutions.

To solve ordinary integro-differential equation, byusing the solution ansatz (see Eq. (3) in Pınar andOzis[2]) and balancing the ansatz, one can easily determinethe solution series as a second order polynomial

υ(ζ ) = g0+g1z(ζ )+g2z2(ζ ) (3)

In z(ζ ) and the coefficientsg0, g1 and g2 the freeparameters can be further determined. Referring to theinitial condition (i.e. initial value) which is in the form ofan exponential distribution in the PBE model, theexpected solution is therefore always exponential.Referring to Case 6 of Table 1 in Pınar andOzis [21], theauxiliary equation can be expressed as:

(dzdζ

)2 = a2z2(ζ )+a6z6(ζ ) (4)

with the solution

z(ζ ) = e

(

− 14LambertW

(

−1a6e(4

√a2(−ζ+ CI))

2a2

)

−√

a2(ζ− CI)

)

(5)

which behaves in an exponential form. Using themethodology in Pınar andOzis [21,22], the parametricgeneral solution of the PBE is given for various casestudies of growth, nucleation and aggregation processesand compared to their analytical solutions that exist in theliterature [25,26,27].

4 Case study 1: Combined Nucleation,Growth and Aggregation

The population balance for a combination of nucleation,growth and aggregation case can be obtained from Eq.1by setting the breakage function and the specific rate ofbreakage to zero i.e.b(x,x′) = 0, Γ (x) = 0. This type ofPBE is typically encountered in MSMPR systems [28,29]and in crystal synthesis [30,31]. For the sake ofcomparison with an analytical solution, the growth andnucleation kernels for the steady-state PBE are assumedto be constants i.e.G(x, t) = G0 = 1, B0(t) = β0 = 1,a(x,x′) = 1. Using the aforementioned conditions, Liao &Hulburt [32] proposed the following analytical solution:

n(ν) = 2n0 exp(−px)I1(x)

x(6)

where

p=

√

1+1

2β0n0G0t2

x=νG0

√

2β0n0G0

n(0) = n0

Using the proposed AEM approach, the analyticalsolution obtained is as follows:

n(x, t) = g0

+g1e(− 1

4LambertW(−1a6e′(4

√a2(Φ))

2a2)+

√a2(Φ))

+g2(e(− 1

4LambertW(−1a6e(4

√a2(Φ))

2a2)+

√a2(Φ))

)2


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where

Φ =−µx+αt+ CI

g0 =−10√

e−100x

g1 =−10√

2e(x−0.1)−2.2

a2 =2000

√e−100x

αµa6 =−0.01tanh(0.1x)

µ = 1.5, α =−0.3, CI =−0.3

0.010 0.012 0.014 0.016 0.018 0.020 0.0220

1

2

3

4

5

analytical AEM solution (this study) analytical solution (Liao & Hulburt, 1976)vo

lum

e de

nsity

n(x

,t)

volume, x

Fig. 1: Comparison of the proposed AEM analytical solutionwith the analytical solution of Liao & Hulburt [27] (Case study1)

As seen in Figure 1, there is a general similarity in thebehavior between the proposed AEM solution and theanalytical solution proposed by Liao & Hulburt [27].However, the slight difference in the solutions isanticipated. This is because even if both the solutions areinherently analytical, the postulations of the derivationofthese solutions differ from one other. As Liao & Hulburt[27] mentioned in their work, the combined aggregation,growth and nucleation was obtained by setting thebreakage function and specific rate of breakage to zeroand additionally the aggregation kernel and growthfunction were both taken to be constants. However, in ourcase, they are supposed to vary with the leading termvolumex.

5 Case study 2: Growth and Aggregation

Similar to the previous case, the PBE for a combinationof growth and aggregation case is obtained by setting thebreakage function and the specific rate of breakage to

zero. Further in this case, the nucleation function is alsoset to zero i.e.B0(t) = 0. In classical modelling approach,the growth and aggregation phenomena are independentlyconsidered. However, applications occur [33] where thegrowth and the aggregation are coupled thus not obeyingthe classical Von Smoluchowski equation. Till date, onlya few analytical solutions have been proposed in literaturefor various combinations of growth and aggregationkernel functions as seen in Table 1.

Table 1: Combinations of growth and aggregation kernelfunctions (Case study 2) obtained from Majumder et al [34] andas found in Ramabhadran et al. [26].

Case G(x, t) a(x,x′)

a 1 100b x x+x′

According to case 2a (see Table 1), the analyticalsolution given by Ramabhadran et al. [26] is given asfollows:

n(x, t) =

M20

M1

1−2Λx0

(

N0−M0M1

)

exp

−M0

M1

x−2Λx0

(

N0−M0M0

)

1−2Λx0

(

N0−M0M1

)

(7)

where

M0 =2N0

2+β0N0t

M1 = N0x0

[

1−2G0

β0N0x0ln

(

22+β0N0t

)]

.

HereΛ = G0β0N0x0

andM0 andM1 are the moments.

The corresponding analytical solution using AEMapproach is obtained as follows:

n(x, t) = g0

+g1e(− 1

4LambertW(−a6e(4

√a2(Φ))

2a2)+

√a2(Φ))

+g2(e(− 1

4LambertW(−a6e(4

√a2(Φ))

2a2)+

√a2(Φ))

)2



where

g0 = 10e−(x−0.2)2

g1 =−e−x

g2 =−e−x

a6 =−10e−x

a2 =tanh

(

x10

)

1000Φ =−µx+αt+ CI

µ = 1.4, α =−0.5, CI = 0.0

A comparison of the two analytical solutions can beseen in Figure 2.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.80

2

4

6

8

analytical AEM solution (this study) analytical solution (Ramabhadran et al, 1976)

volume, x

volu

me

dens

ity n

(x,t)

Fig. 2: Comparison of the proposed AEM analytical solutionwith the analytical solution of Ramabhadran et al. [26] (Casestudy 2a as mentioned in Table 1)

According to case 2b (see Table 1), the analyticalsolution given by Ramabhadran et al. [26] is as follows:

n(x, t) =M0

x√

1− M0N0

exp

[

−M0

M1

(

2N0

M0−1

)

x

]

I1

(

2xN0

M1

√

1−M0

N0

)

(8)

where

M0 = N0exp

[

β0N0x0

G0(1−exp(G0t))

]

M1 = N0x0exp(β0t)

Note thatI1 is the modified Bessel function of first kind oforder one.


n(x, t) =13+g1e

(− 14LambertW(−

a6e(4√

a2(Ψ ))

2a2)+

√a2(Ψ ))

+6g2

1(e(− 1

4LambertW(−a6e(4

√a2(Ψ ))

2a2)+

√a2(Ψ ))

)2

18x+1(9)

where

g1 = 10e−7.84x2

a2 =−10e−x

a6 = 25tanh(1000)

Ψ =−µx−

(

6g21−

6g21

18x+1

)

(18x+1)µt

g21

+ CI

µ = 2.8, α = 3.5, CI =−4.0

A comparison of the two analytical solutions can beseen in Figure 3.

0.00 0.02 0.04 0.06 0.08 0.10 0.120

1

2

3

4

5

6

7

analytical AEM solution (this study) analytical solution (Ramabhadran et al, 1976)

volume, x

volu

me

dens

ity n

(x,t)

Fig. 3: Comparison of the proposed AEM analytical solutionwith the analytical solution of Ramabhadran at al. [26] (Casestudy 2b as mentioned in Table 1)

It is worth noting that the AEM solutions of cases 2aand 2b, similar to case 1, preserve the general trend of theexponential distribution and the slight difference is againbased on the derivation of the analytical solutions based ondistinct assumptions.




6 Case study 3: Nucleation and Growth

Similar to the previous cases, the PBE for a combinationof nucleation and growth is obtained by setting thebreakage function and the specific rate of breakage tozero. However in this case, instead of the nucleationfunction, the aggregation kernel is set to zero i.e.a(x,x′) = 0. Studies related to combined nucleation andgrowth is undergoing a growing importance innanotechnology [35]. Several analytical solutions havebeen proposed in literature for various combinations ofnucleation and growth phenomena. From these, two caseswith combinations of growth and nucleation kernelfunctions (see Table 2) are considered.

Table 2: Combinations of growth and nucleation kernels (Casestudy 3) as obtained from Kumar & Ramkrishna [25].

Case G(x, t) B0(t)a 1 105

b 1 10

According to case 3a (see Table 2), Kumar &Ramkrishna [25] proposed the following analyticalsolution:

n(x, t) = n0(x−G(t))

+N0,n

σ0

[

exp

(

−xlow

x0,n

)

−exp

(

−x

x0,n

)]

(10)where

xlow = max(x1,x−σ0t).

In Eq. (10), n0(x) is the initial number density andN0,n represents the nucleation rate.x0,n and σ0 indicateparameters related to exponentially distributed nucleationrate and growth rate function respectively.


n(x, t) = g0

+g1e(− 1

4LambertW(−a6e(4

√a2(Φ))

2a2)+

√a2(Φ))

+g2(e(− 1

4LambertW(−a6e(4

√a2(Φ))

2a2)+

√a2(Φ))

)2

where

g0 = 10e(−(x+0.6)2

g1 =−e−x

g2 =−e−x

a2 =tanh( x

10)

1000a6 =−10e−x

Φ =−µx+αt+ CI

µ = 1.6, α =−0.8, CI =−3.0

0.0 0.2 0.4 0.6 0.8 1.02

4

6

8

10

12

volume, x

volu

me

dens

ity n

(x,t)

analytical AEM solution (this study) analytical solution (Kumar & Ramkrishna, 1997)

Fig. 4: Comparison of the proposed AEM analytical solutionwith the analytical solution of Kumar & Ramkrishna [25] (Casestudy 3a as mentioned in Table 2)

According to case 3b (see Table 2), Kumar &Ramkrishna [25] proposed the following analyticalsolution:

n(x, t) = n0(xe−G0t)e−G0t

+N0,n

σ0

[

exp

(

−xlow

x0,n

)

−exp

(

−x

x0,n

)]

(11)

where

xlow = max(x1,xe−G0t)


n(x, t) = g0

+g1e(− 1

4LambertW(−a6e(4

√a2(Φ))

2a2)+

√a2(Φ))

+g2(e(− 1

4LambertW(−a6e(4

√a2(Φ))

2a2)+

√a2(Φ))

)2

(12)

where

g0 = 10e(−(x+0.62)2

g1 =−e−x−0.02

g2 =−e−x−0.02

a2 =tanh(x+ 0.02

10 )

1000a6 =−10e−x−0.02

Φ =−µx+αt+ CI

µ = 1.6, α =−0.8, CI =−3.0

The proposed AEM solutions related to cases 3a and3b are compared with their corresponding available



0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

12

analytical AEM solution (this study) analytical solution (Kumar & Ramkrishna, 1997)

volume, x

volu

me

dens

ity n

(x,t)

Fig. 5: Comparison of the proposed AEM analytical solutionwith the analytical solution of Kumar & Ramkrishna [25] (Casestudy 3b as mentioned in Table 2)

analytical solutions, as seen in Figures 4 and 5respectively. As can be clearly observed, the solutionsproposed using AEM and suggested by Kumar &Ramkrishna [25] match very well. The slight differencesthat exist between the solutions are again based on thederivation of the analytical solutions based on distinctassumptions. In the literature, several analyticalformulations are available to represent various particulateprocesses based on various hypothesis and/orassumptions. But, in principle, such analyticalformulations should be equivalent and can essentially betransformed from one to the other through certainequations. Although this proposal is theoretically correct,but in practice this is not observable in many casesbecause the population density versus size functiontypically used to model particulate processes and thepopulation, in general, is used as the distributed variablerather than mass/or particle volume. For example, inseveral PBMs linear size is used as an independentvariable rather than particle volume for simplicity. But inparticulate processes, when aggregation (agglomeration)or breakage is more important the particle volume mustbe used rather than size (say, diameter). It is important tonote that although the volume (or mass) is conserved inthese processes, linear size is not conserved. Figure 6shows the volume conservation behavior of this approachusing case 2a (as an example).µ is the volume coefficientas we use transformation which depends on both volumeand time. As seen in Figure 5, whenµ changes the curvedoes not change thus indicating volume (mass)conservation. The AEM solution proposed in this study isfound to be compatible with the various analyticalsolutions obtained from the literature, and thus adds to thebelief that AEM is a more robust approach.

0.010 0.012 0.014 0.016 0.018 0.0200

2

4

6

volume, x

volu

me

dens

ity n

(x,t)

= 55 = 60

Fig. 6: Numerical verification of volume (mass) conservationusing AEM approach

7 Conclusions

In this study, an effective analytical technique isimplemented to solve PBEs involving simultaneousparticulate growth, nucleation and aggregation by makinguse of the appropriate solution(s) of associatedcomplementary equation via auxiliary equations.Travelling wave solutions of the complementary equationof the nonlinear partial integro-differential equation withappropriately chosen parameters is taken to be analogousto the description of the dynamic behavior of theparticulate processes of a PBE. Using carefully chosenparameters, the AEM solution is able to reproduce theexpected behavior of various particulate conditions and iscompatible with the analytical solutions proposed in theliterature.

Acknowledgement

The manuscript has been prepared based on the workpresented in ICAAAM 2013 held at Istanbul, Turkey. ADgratefully acknowledges TurgutOzis and Pınar Dundarfor granting a brief research stay at the Department ofMathematics in Ege University (Turkey) during April2013.

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Zehra Pınar receivedher B.S. degree in AppliedMathematics from EgeUniversity, zmir, Turkey in2007 and her M.S. and PhDdegrees also from the sameuniversity in 2009 and 2013,respectively. In January 2014,she joined Namık KemalUniversity, Tekirdag, Turkey

as an Assistant Professor. Current research interestsinclude analytically-approximate and numerical methodsfor nonlinear problems in physics and engineering.

Abhishek Dutta obtainedhis Bachelors in ChemicalEngineering from Universityof Madras in 2002 andMasters in Food Technologyand Biochemical Engineeringin 2004 from JadavpurUniversity, both fromIndia. After a brief industrialstint, he undertook a PhD in

Chemical Engineering and a subsequent post-doctoralresearch in BIOMATH, both at Ghent University,Belgium from 2006-2011. Presently he is working as afull-time guest Lector (voltijds gastdocent) at the Facultyof Engineering Technology, KU Leuven (Campus GroepT Leuven), Belgium. Current research interests includemathematical modeling using multiscale and involvingmultiphase systems.

Guido Beny obtainedhis PhD in Biochemistry fromVrije Universiteit Brussel inthe field of Bacterial Geneticsin 1984. He presently worksas a docent at the Facultyof Engineering Technology,KU Leuven (CampusGroup T Leuven) and is theUnit Manager of Chemical

Engineering at that campus mainly involved in education.He has active research interest in the understanding offundamental phenomena in (bio)chemical systems.



Turgut Ozis obtained hisB.S. degree in Mathematicsfrom Ege University, Izmir,Turkey, in 1973. He receivedhis M.S. and Ph.D. degreesfrom Brunel University,London, U.K. on free andmoving boundary problems in1981. In late 1982, he movedto what was then Inonu

University, Malatya, Turkey, and worked as the Head ofthe Mathematics department until 1991. After that he waspersuaded to accept an Associate Professorship positionin the Science Department of Ege University. In 1992, hewas appointed as a Professor of Applied Mathematics atEge University, since he had been the Head of theApplied Mathematics Group. His research interestsinclude numerical methods for partial differentialequations (both linear and nonlinear), free boundaryproblems, reaction diffusion problems, numerical heattransfer, two-point boundary value problems for ordinarydifferential equations, and exact approximate methods fornonlinear systems. Prof. Ozis is a member of the editorialboards of three international journal.



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