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A UNIFIED INTRODUCTIONTO ORDINARY DIFFERENTIALEQUATIONSCarl V. Lutzer PhD aa Department of Mathematics and Statistics,Rochester Institute of Technology, Rochester,NY, 14623-5603, USA E-mail: http://www.rit.edu/∼cv#sma/Published online: 13 Aug 2007.
To cite this article: Carl V. Lutzer PhD (2006) A UNIFIED INTRODUCTION TOORDINARY DIFFERENTIAL EQUATIONS, PRIMUS: Problems, Resources, and Issues inMathematics Undergraduate Studies, 16:4, 363-375
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Lutzer A Unified Int roducti on to Ordinar y Different ial Equations
A UNIFIED INTRODUCTION TOORDINARY DIFFERENTIAL
EQUATIONS
Carl V. Lutzer
ADDR ESS : Depar tm ent of Math emati cs and Statisti cs, Rochester Insti t ute of Techn ology, Rochest er NY 1462:3-5603 USA. cv£sma@rit . edu.http: //www .r i t .edu/~cv£sma/ .
ABSTRACT: This article describ es how a presentation from the pointof view of differential operators can be used to (pa rt ially) unify t hemyriad tec hniques in an introd uctory course in ordina ry differentialeq uations by providing st udents wit h a powerful, flexi ble pa rad igmt hat extends into (or from) linear algebra.
KEY\VORDS: Mathematics educat ion, ordina ry differential equations,operators.
1 INTRODUCTION
St udents taking their first course in ordina ry different ial equations (ODE)enco unter t he disconcerting fact t hat seemingly min or differences in equat ions' form require dr am ati cally different techniques. At the end of anint roductory course, t hey are often left with a nagging uneasiness about thesubject becau se it seems to lack any hint of rhyme or reason . Of course,t here is rhym e and reason to the subject, but many students walk awaywith only mechani ca l technique and lit tl e if any rea l understanding. Whendiscussing the matter with Dr. George Thurston , he opined that (we paraphrase),
T here seems to be a certain pedagogical momentum that drawsus away from the orig ina l moti vat ion and und erstanding of ato pic-away from the why of th ings , toward t he what and howof it .
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In t hat langu age, man y und ergr aduates are t au ght the how of ODE but nott he why of it . In short , t his art icle is abo ut presenting a signifi cant porti onof an introductory course in ord inary differenti al equa t ions in a coherent,unifi ed way that provid es st udents with both t he "what" and t he "why" ofintroduct ory ODE. The suggested paradigm will be a pplied to t hree spec ificcases:
F irst , conside r t he case when the charac te rist ic polyn omi al of m y" +rY' + ky = 0 has a double root , say A. We know t hat y](t) = e At so lvest he eq uation, bu t where do we get a second solut ion? T hinking from thestandpoint of t he novice, what would make us t hink t hat multiplyin g by twould resul t in a second solut ion? This is very closely rela te d to t he nextqu esti on .
Second , in t he tec hnique we call undetermined coefficients, we makean ed ucated guess abo ut t he form of the solut ion based on t he form oft he driving function; bu t when our guess is a solut ion of t he associatedhom ogeneous equation we have to multiply by t . It is easy eno ugh to checkt hat it works in pr acti ce, but why does it work?
T hird , conside r t he t echnique we call variation of param eter-so In bri ef,we begin our sea rch for a par ti cul ar solut ion to m y" + r Y' + ky = g byfinding two (linearly ind ep endent ) solut ions to t he homogenous version oft he problem ,my" + r Y' + k y = o. Let us nam e t hese solutions w, a nd'1/;2. In an introductory course, teachers ofte n spend t ime belaboring t hepo int that any linear combinat ion of t hese two fun cti ons will also solve t hehom ogeneou s ODE, so we can work with the fun cti on Cl tlJ] + c2t1J2 , whereCI a nd C2 a re any two sca lars . T his form of t he so-called compleme ntarysolution allows us to achieve two ini ti al conditions. So far so goo d.
T he ped agogical problem comes just afte r we say t hat any linear combination of l/J I a ndl/J2 will solve the homoqeneous ODE. Thinkin g from thestandpoint of t he novice, knowin g that t hese two fun cti ons are solut ions tot he homoqeneous equation, wh at would make us believe t hat c l l/J] + c2 l/J2
could solve the nonhomogeneous equation if we allow c] and C2 to vary inti me? We j ust sa id t hat any such comb ination would solve t he hom ogeneousversion of t he equation! Of course, we can ver ify t hat t he t echnique works,but does it follow naturally from simpler ideas? Does it m ake sens e?
As t he read er might ant icipate , t he variation of pa ramet ers tec hniquedoes follow from simpler ideas; and it happens in a very natural way, t husavo id ing t he qu estion , "W hat would make us think. . . ?" T he spec ifics oft he development are demonstrated in Section 5 of t his art icle , immedi atelyafter t he question of "why?" associa ted with undetermined coefficients isanswe red in Section 4. But before t hat , Section :} dem onst rates how t he
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Lu tz er A Unified In t roduction to Ordinary Differ ential Equation s
ty' +y a nd
d(d ) " ,- -(ty ) = t y + 2.11 .di di.
t eAt so lution a lso arises naturally when t he characterist ic equation has ado ub le root. '\, and Section 2 ela borates on t he paradigm t ha t t ies it a lltoget her.
2 A UNIFYING IDEA
Answers to t he qu esti on s from the previ ou s sect ion arise na t ur all y whe n wespeak in t he langu age of linear operators. Lin ear ope ra tors are addressedin 1II0~t 1II0d em books for int roductory courses, but they often appear as aparti cul ar topi c or a s ub-to pic (see [1, p . 193] or [2, p . 14;~]) . I3y contrast ,we propose to ado pt lin ea r operators as "a way of life" (to use a ph ra se ofDr. Russell I3rown ). At t he point in the course when second orde r, lineareq ua t ions a re introduced , st ude nts sh ould be reacquainted with differentialoperators and t he idea that they "act " on every thing to the right -c -hore wesay "reacquainted" becau se such operators are not beyond their ex pe rience.Indeed , students who are t aking a co urse in ODE should alread y be familiarwit h t he product rul e , which we write as
ddt(t y)
d'2ilt 2 (ty )
St ud en ts sho uld also be show n tha t d ifferenti al operators do not commute ingelieraI, bu t t ha t they do when t he coe fficients a re constant. Once t he ideaaw l notat ion of different ial operators established , it is very easy to definewh at it. means for a n ODE to be linear . We say t hat a d ifferential eq uat io nis linear if it ca n ln- wri tt en as
/I f k'
2: lld t ) (~tkY = y(t).k=O
Now suppos ing t hat student s a rc comforta ble wri ting .II" + by' + cy = () as
((j'2 II )- ') + /J- + (' .II = 0 ,dt- di
(o r a t lea st. that they 're comfortable watching som eone else do it ) we solvet he seco nd orde r eq uation by fact oring t he linear op erator on the left-h andside . In t he case of constant coe fficients , t his is as easy as factoring apolyn omial. For exam ple, t he equa t ion .II" + 8.11' + 15.11 = () ca n be wri ttena~
(P d )(~t2 + 8 dt + 15 .II 0, (1)
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and t he operator on the left-hand side ca n be factored so that t he equat ionbecom es
o. (2)
Writ ten t his way, we ca n und erst and t he left-hand side of t he equat ion astwo actions performed in order: first (-it + 5) acts on y , and t hen (-it + :3)acts on t he result to prod uce t he zero funct ion. T his would certainly happenif t he first step of t he process resul t ed in t he zero function-i.e., one wayfor t he fun cti on y to solve (2) is for
(~ +5) y = 0=} y' = - 5y =} y = c- 5t.
elt
The first-order linear operators (-it + :3) and (-it + 5) commute since t heyhave const ant coefficients, so we could ju st as easily have wri t ten (1) as
(~ +5) (~ + 3) Y = 0dt di '
from which t he same argument lead s us to t he solut ion y = c- 3 t . Nowwith two linearl y ind ependent solut ions in hand , we ca n writ e t he generalsolution to t he hom ogeneous equation as
Of course, t his is exactly t he same form of t he genera l solution t hat ouefinds in any int roductory text on OD E (for example, [1, p.lG9] or [2, p.l :38]);but presenting t he solution technique from t he factored-operator poin t ofview helps to make later material seem more natural , which we sha ll seein t he following sect ions, and helps the st ude nt by reinforcing (or perhap sintroducing) an import an t idea in mathemati cs: in or der to solve a big orcom plica te d probl em , we solve several smaller , eas ier probl ems. In t his caset he complicated problem was t he single second-or der equat ion which webroke into two first -o rde r equations , each of which was easy to solve.
We pau se to rem ark t hat st udents are also taught to factor linear operators (in t he guise of matrices) in courses on linear algebra. In particul ar ,t he LU decomposition gives us a very fast way to solve the equat ion Ax = bover and over again wit h di fferent right-hand sides (see [3, p.154]). In brief,t he technique is to factor A = LU, where L is a lower t riang ular matrixand U is an upper triangular mat rix. T hen we solve LU:l; = b in two st eps:
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(1) find a vector v t hat solves Lv = b, and (2) find a vector :1; t hat solvesUx = v.
Ax = L Ux = b.'-..,-'
= v
(4)
T he ideas of t he next sect ion are also closely linked to ideas in linear a lgebra,a brief discussion of which can be found in App endix B.
3 DOUBLE ROOTS OFCHARACTERISTIC POLYNOMIALS
On ce we have factored our differential opera to r, we can use t he two-stepsolut ion techn ique from linear algebra for use in solving OD Es. Let usconsider t he case when the linear opera tor is a perfect square,
Ly = (~ - A) (~ - >.) u = 0dt dt '
'-v---''-v---'L 2 L,
(5)
where we have la beled the two twin opera tors as L 1 and L2 (even thought hey are the same) becau se it will make our discussion eas ier. In this case,t he technique from t he previous sect ion would lead us to the solut ion eAt,
bu t we would have no way to find a second solut ion. T his is where t he ideaof itera ted probl em solving comes in handy. We begin by findi ng a funct ionv such that L2 v = 0, and then solve the equat ion L 1y = V. So
Ly = £ 2 L I Y = O.'-..,-'
=v
The equat ion £ 2V = Uk - A) Ii = VI - Au = 0 is solved by u(t) = eAt.Solving the second equation is a lmost as easy. We make use of an integratingfactor to write
yl - Ay
yle- At _ Ae- Aty
J:t (y e-At)
dt
ye- At
1
J1 dt
t + C.
Taking C = 0 to make our lives easy, we arrive at y(t ) = teAt, which is anew solut ion to our equation.
T he impor tan t pedagogical aspect of this method is that the secondsolut ion to (5) is neit her handed to the students ex cathedra nor unear th ed
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by guess-and-check methods in an accident al fashion. Instead , we discovert he solution natura lly by looking at t he problem in t he righ t way. T heso lution m akes sense from t he iterated-op erators point of view, so now t henovice knows both what happens and why.
4 UNDETERMINED COEFFICIENTS
When teaching t he method of undetermined coefficients (ed ucated guessing), we run into probl em s when our guess is annihilated by the differen ti alope rator. For exam ple, when t ry ing to solve
y" - 4y' + :3y = 7et, (6)
we would guess t hat a parti cul ar solut ion to t he equation is ¢(t ) = A et , bu t¢" - 4¢ ' + 3¢ = 0, so no such fun ction could ever work. What do we do?Mul tiply our guess by t . Why? T hat is a mystery for many st ude nts, bu tt he it era t ed solut ion method makes t his idea natu ral. Let us rewri te (6) as
(d ) (d .) tdt - 3 dt - 1 Y = 7e .
'---v--''---v--'!lI2 !III
Now we see that A et cannot be a solut ion becau se it is annihila te d by M« .It would be nice if we could get lIf[ to creat e et instead of annihila te it , fort hen we'd be left wit h a collection of et after 1112 acted (et is not annihilatedby lI f 2 ) . T hat is exactl y what we wan t to see.
T he prev ious section shows exact ly how t his ca n happen. When we werelooking for a second solution to (5), we found t hat L [ (t eAt) = eAt so t hatL2L[ t eAt = L 2eM = O. T hat is , L[ created e M by acting on t e/":
In t he current context , t he key to finding a par ti cul ar so lut ion is to beginwit h t et , so t hat the action of Jl f [ will resul t in et . Th at 's why we multiplyby t .
5 VARIATION OF PARAMETERS
T he technique we ca ll variat ion of parameters also ar ises from an iterat edsolution pro cess. We begin with a par ti cul ar example in or der to introducet he technique.
Example: Find a part icula r solution to y + Sf;+ 15y = et .
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We begin by rewriting the equation as
and t hen pass thro ugh the two-s te p process.
St ep 1 (so lve L2v = e' ]: The equation v' + 3vusing an integrating factor:
el is eas ily solved by
e3lv' + 3e3t v e4t
J~ (e3lv) dt = Je4tdt
e31v O.25e4 1 + kl ·
'vVe choose k, = 0 to make the remaining algebra nice, so v = O.25et .
S tep 2 (so lve L l y = u}: As in t he previous step, we also solve the second equation by way of an int egrating factor.
e51iJ + 5e51y
J:t (e5t y) dt =
Also as before, we take k2 = 0 in order to simplify the algebra of our steps.This leaves us wit h y = i4 et as our par ti cular solut ion. It is easy to checkthat this function solves the ODE, but where is the variat ion of parameterstechnique? We see it more clearly by considering the genera l case.
Example: Find a part icular solution to y" + by' + cy = g(t ).
We begin by factoring the differenti al opera to r on th e left-hand side of th eequation so that the equation becomes
( : t -AI) ( : t - A2)y=g(t)."-v----""-v----"
L 2 LI
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As before, we begin by solving the equation L 2v = V i - A\ V = y (t ) with anintegrating factor.
V
e- >\l tg (t )
Je- >' d g (t ) dt
Je- >' l tg (t ) dt
e>' l t {J e- >' l tg (t ) dt} .
Here, the integral has been delimi ted with large braces in order to help uskeep track of the fact that it is a function , \ even though we have not found aclosed-form ant iderivative. A very imp ort ant fact about this functi on comesfrom t he Fundamental Theorem of Calc ulus:
The second step in the pro cess is to solve t he equat ion y' - A2Y = V , whichwe also do with an integra ting facto r.
e>' d {J e- >' l tg (t ) dt}
e(>' 1- >'2) t {J e- >' Jly (t ) dt}
e(>' 1- >'2) t {J e- >' l tg (t ) dt}
Je(>'1- >'2)t {J e- >' l tg (t ) dt} dt (7)
The right-hand side of (7) can be cleaned up with an integration by parts.Because we're working in genera lity, we have to consider both the case whenAl = A2 and the case when Al f= A2. The same technique works in bothcases, but the results look different.
Case 1, when Al = A2: In t his case, e(>'1- >'2) t = 1, so (7) becomes
(8)
1Of co urse, t he indefinite integral is reall y a family of fun cti ons. Here we a re ass umingthat a represent ative member of t he family has been chosen, ju st as we did in t he previou sexample by set t ing k l = O.
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Using t he common udv notation, we set
II = {f e- Atly(t ) dt }
du = e-Atl g(t) dt
T his a llows us to rewri te (8) as
dv = 1 dt
v= t.
from which we conclude that our particular solut ion is
(9)
Case 2, when)'1 and A2 are distin ct: Using th e common udv not ation, weset
II = {f e- A11 y( t) dt}
du = e- A1tg(t ) dt
T his allows us to rewri te (7) as
from which we conclude t hat our par ti cular solut ion is
(10)
\Vhen AI and A2 are complex conj ugates, Eul er 's formula, e iO = cos B+i sin () ,can be used to rewrite th e expression in term s of trigonometric functionsinst ead of complex exponent ia ls.
Both (9) and (10) have th e form y = CI( t) 'l/J1 + C2(t )l/J2, where l/JI andl/J2 are (linearly independ ent) solut ions to th e homogeneous equat ion-soth e two-step solut ion method has led us naturally to th e idea of lettingt he coefficients of the linear combina t ion var y with time in order to find aparti cular solut ion!
Let us rem ark that both (9) and (10) agree with th e standa rd formulat hat st udents find in their text (see [1, p. 222] or [2, p.189]):
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where W (¢2, ¢ d is t he Wron skian of t he fund amental so lut ion set. However, we a rrived at t he formula wit ho ut any ment ion of t he Wronskian. (Ofcourse, t h is lead s to t he qu esti on of what place t he Wron ski an should playin an int roductory course, but that's a to pic for ano t her note. In any event,formulas (9) a nd (10) can be used to assert it s importance.)
CONCLUSION
Using t he langu age and notati on of differen ti al operators in an introductoryclass has severa l benefits , t ho ugh there is some "leg wor k" that needs to bedone up fron t. By factoring linea r operators (in the constant coefficient case)and employing a two-step solut ion method , t he "second" solut ion arises natur all y when t he characterist ic polynomial has a double root . T he exact sametwo-step method allows us to underst and why multiplicati on by t happenswhen t he characterist ic equation has a double root , and leads us di rectl y tot he varia t ion of parameters technique! T hese t hings are now all relat ed by asing le idea , so t he novice can underst and eleme ntary differential equationsas a single (somew hat) unified subjec t inst ead of 100 lit tl e ones. Further,the two-step method as seen in an introductory course in ODE reinforces (orintroduces) import ant ideas and tec hniques t hat st ude nts will see in othercourses. The con fluence of t hese benefits gives st udents a greater sense ofstability, a nd more importan tl y, it provides them wit h some understand ingof why t hings happen , not just that t hey happen .
APPENDIX A
At t he onset of th is article, we asked whether t here is a simple, naturalreason to expect t e/" as a second solut ion to
(11)
An alternative method of deriving the form of the second solut ion is to a pproximate t he ODE by a family of simila r ODEs, eac h wit h two distinctroot s of it s characterist ic polyn omial , that converge to (11) in some paramete r . For example, if we write (11) as
(12)
it migh t occ ur to us to approximate it as
(13)
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which is solved by eAt , eA1t , and any linear combination thereof. In particula r, the OD E is solved by
(14)
Now as we let )'1 ----t '\ , equation ( la) becomes (12) and
(15)
since t he right-hand side of (14) is ju st t he difference quotient of t he exponenti a l function at ,\ (t hinking of t as a par am et er and ,\ as the variable) .
While this argument is sound, it 's not as natural and st ra ight -forwardas th e two-ste p method - at least not to th e novice. The method out linedby (11)-( 15) begins not with an approximat ion of th e solut ion, but with anapproximation (pert urbation) of the equation it self. That is a very sophist icated way of looking at th e solut ion pro cess, and one that most st udentsdon 't see anywhere else in their und ergraduate st udies. Beyond t hat fact,t he par ti cular form of t he approx imating family of equations- while exact lythe right thing to do in hindsight- seems to come out of the blue, and understanding (15) requires st udents to twist around t heir po int of view sot hat t is a par am eter (instead of the independent variab le) and ,\ is t he indep end ent variable (instead of a parameter ) for t he different iation process.Tone of these steps are particularl y hard , and we are not ar guing that t he
above derivation is a bad one. We contend only that the meth od out linedby (11)-(15) lacks motivation from t he novice's point of view, and so mayseem more magical tha n math emati cal.
AP PENDIX B
In Secti on 4, we used t he idea that a different ial operator could both annihilate et and create e': This was done in th e case when the cha rac te rist icpo lynomial had a double root , and here we remark that this idea (like th etwo-step method itself ) has its origins in linear algebra . While an in-depthdiscussion is beyond the scope of the cur rent art icle (and certainly beyondthe scope of an introductory course in ODE! ), let us briefly indi cate th ebas ic ideas involved.
Any second-order, linear equation with constant coefficients can be written as a first- order matrix equation by int roducing an auxilia ry varia ble. Forexample, y" + by' +cy = 0 can be written as y" = - by' j a-cyja. Int roducing
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(16)
the a uxilia ry variable z = y', we ca n write t his as
(~:) [O~ _1~ ] (~) ."--v--'
A
When the characterist ic pol yn omial of the ODE had a double root, thecoe fficient matrix , A, is nondiagon alizable. At a fund am ental level , t hisnondiagon ali zability is t he problem . So what do we know, at a fund am ent al level , a bo ut such matrices? Most importantly in this context, we knowt hat t hey (like t he ir diagonalizable cousins) are solutions of their own characterist ic equati ons- t his is t he famous Cayley-Ham ilton Theorem (see [;),p .474]) , whi ch says that an ti x n matrix A with k distinct eigenva lues,AI , . . . , AI;, satisfies
I;
II (A - AjI)ui = 0,j = 1
wh ere t he zero on t he right-hand side of (Hi) is the zero matrix. T hat is,for any vector :c E C" ,
This makes perfect sense when A is di agonali zabl e, because the alge bra icmultiplicity of each eigenvalue is the sa me as it s geo metric multiplicity.That is, (A - AjI) annihila te s aj dimensions, so IT~·=I (A - AjI) annihila tes
L~=I elj = n dimension s, Notice that we only needed to use on e of eac hfactor in order to achieve the result. However we need th e ot her factors wh enA is not di agon ali zabl e, becau se at least one eigenva lue is deficien t (i.e., it sgeometric multiplicity is strict ly less than it s alge bra ic mul tiplicity ).
Let us suppose t hat AI is deficient . Then (A - AlI) annihila tes lessthan a l dimensions, so IT~=1 (A - AjI) annihilates less than n dimensions,whi ch is why t he extra factors of (A - AII) are required to make (lG) true.However, let us continue to focus on t he acti on of the "first" factor: equation(16) bears out becau se this first factor not onl y annihilates some of t he a l
dimen sion s that we need it t o, but also rotat es other vectors into 'its kern el.This a llows the next factor of (A - )'1 I) t o annihila te more dimensions.The second facto r also rotates new vectors in to t he kernel , so that t he thirdfactor annihila tes even more dimensions- annihilate & rotate, annihila te& rot ate, annihila te & rota te. . . The net result is t hat the a l fact ors of(A - AII) annihilate a I dimension s.
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Lutzer A Unified Introducti on to Ordinary Differential Eq uations
This is th e intuitive idea that we're using to solve (6) . Instead of focusingon the fact that All annihilates e'', we look for th e "vector" that it "rotates"into its own "kernel."
ACKNOWLEDGMENTS
Thanks are exte nded to George Thurston (RIT Physics Department),Bernard Brooks and H. T. Goodwill (RIT Department of Mathematics& Statistics), and Russell Brown (University of Kentucky Department ofMathematics) for helpful discussions about this and many other subjects.The author also wishes to not e that , though he has used [1] and [2] by wayof contrast , he has great admirat ion for th ese texts and uses them often asreference for his own teaching.
REFERENCES
1. Borrelli , R . and C. Coleman, C. 2004. Differential Equations-AMod eLing Perspect ive, Second Ed it io n. New York: John Wiley & Sons.
2. Boyce, W . and R. DiPrima. 2005 . ELementary DifferentiaL Equations,
Eighth Edition. New York : John Wiley & Sons .3. Anton, H. and R. Busby. 2003. Conternpom ry Linear ALgebm. New
York: Jo hn Wiley & Sons.
BIOGRAPHICAL SKETCH
Carl Lutzer is an Assistant Professor of Mathematics at t he Rochest er Inst it ute of Technology, and was selected for inclu sion in the Who 's WhoAmong Ameri ca 's Teachers in both 200:3 and 2004 . He was a finalist forRIT's Richard and Virginia Eisen hart Provost 's Award for Excellence inTeaching in both 2002 and 2003 , and was a 2000-2001 Exxonlvlobil ProjectNExT Fellow . He earned his PhD from the University of Kent ucky underthe direction of Dr. Peter Hislop. His mathematical resea rch interest s tendto lie in pa rt ial different ial equations, physics, and mathematical biology.In addition to mathem ati cs and teaching, he enjoys writ ing fict ion, fencing(sabre), and playing with his adorable kids.
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