RESEARCH INTERESTS
My primary interest is in the construction of exact solutions of partial differential equations (PDEs).  Recent work with linear PDEs has seen the technique of Darboux transformations extremely fruitful in identifying new exactly solvable parabolic and hyperbolic PDEs.  In the area of nonlinear PDEs, several analytical methods, including symmetry analysis and point and contact transformations have been utilized in obtaining new exact solutions to physically important equations such as the nonlinear heat equation with a source term and the governing equations modeling the flow of granular materials.

PRESENT STUDENTS
Undergraduate
Luis Suazo - Symmetry analysis and first order compatibility of 2D reaction diffusion equations

Graduate
Ashley Embry - Traveling waves solutions for a cross diffusion model
Ethan Hereth - First order compatibility of reaction-diffusion systems
Sydney Marks - Nonclassical potential symmetries for a wave equations system
Jason Torrence - On a class of solutions for highly frictional rat-holes

PAST STUDENTS
Undergraduate
Yousef Abbasi, Trey Bass, Luke Brown, Chad Fendt, Bryan Gipson, Joel Harris, Ethan Hereth, Sarah Jacobs, Garth Johnson, Brandon Lindley, Casey Milford, Jesse Pratt, Katie Reynolds, Bode Sule, Jason Torrence, Brandon Willis and Charity Washam .

Graduate
Regan Beckham

CONFERENCES

AMS - MAA Joint Meeting, January 2000 “A link between solutions of linear partial differential equations” AMS - Southeastern Sectional Meeting, April 2000 , “Symmetries of a higher order diffusion equation” MAA OK-AR Meeting, April 2001,  “Darboux transformations of the Schrödinger equation " AMS - MAA Joint Meeting, January 2001 “Darboux transformations for linear partial differential equations” AAS Meeting - “Darboux transformations of the Schrödinger equation” AMS - MAA Joint Meeting, January 2002 “An nth order Darboux transformation for the one-dimensional time dependent Schrodinger equation” MAA OK-AR Meeting, April 2002,  “An nth order Darboux transformation for the one-dimensional time dependent Schrodinger equation” AMS - MAA Joint Meeting, January 2003 “Charpit's method and symmetry analysis” MAA OK-AR Meeting, April 2003, “Charpit's method and symmetry analysis” AMS - MAA Joint Meeting, January 2004,  “Nonclassical symmetry analysis and compatibility. MAA OK-AR Meeting, April 2004,  “Generalized compatibility for the two-dimensional diffusion equation with a source term AMS - MAA Joint Meeting, January 2005,  “Generalized compatibility for the two-dimensional diffusion equation with a source term

1.  Group Properties of a Monge-Ampere Equation
     in "Computational and Applied Differential Equations II - Differential Equations", pp 107-116
     W. F. Ames and P.J. van der Houwen, Eds., North-Holland Publishers
2.   Motion of Planar Bubble Boundaries, (with R. A. Wentzell and G. J. Lastman),
     Int. J. Nonlinear Mech. 25(1), pp 87-97 (1990),
3.   Group properties of  u_xx +  u_y^mu_yy   =  F(u),
     Int. J. Nonlinear Mech. 26(5), pp. 619-629 (1991)
4.   Nonclassical symmetry solutions and the methods of Bluman-Cole and Clarkson-Kruskal,
     (with P. Broadbridge and J. M. Hill), J. Math. Phys., 34(10)  pp. 4692-4703 (1993)
5.   Point and Potential Symmetries of Partial Differential equations,
     J. Math. Anal. Appl.,183(1) pp. 167-180 (1994)
6.   Nonclassical symmetries of the linear heat equation with a nonlinear source term,
     (with J. M. Hill and P. Broadbridge) I. M. A. J. Appl. Math., 52 pp. 1-24 (1994)
7.   On axially symmetric deformations of perfectly elastic compressible materials,
     (with J. M. Hill)  Q. J. Mech. Appl. Math., 49(1)  pp. 19-28 (1994)
8.   Nonclassical symmetries of the nonlinear heat equation with a source,
     (with J. M. Hill)  Studies Appl. Math., 94  pp. 21-39 (1995)
9.   New families of exact solution for finitely deformed incompressible materials
     (with J. M. Hill)  I. M. A. J. Appl. Math., 54 pp. 109-123 (1995)
10.   Nonclassical solutions are non-existent for the heat equation and rare for the nonlinear diffusion equation
      (with J. Goard and P. Broadbridge) J. Math. Anal. Appl., 202 pp. 259-279 (1996)
11.   Transformations and equation reductions in finite elasticity. I. Plane strain deformations
      (with J. M. Hill) Math. Mech. Solids, 1(2) pp.155-175 (1996)
12.   Transformations and equation reductions in finite elasticity. II. Plane stress and axially symmetric deformations
      (with J. M. Hill), Math. Mech. Solids, 1(2) pp.177-192 (1996)
13.   The integrable nonlinear degenerate equation  u_t = (u/u_x)_x and its relatives,
      (with J. Goard and P. Broadbridge) ZAMP 47 pp. 926-942 (1996)

14.   Small deformations superimposed upon the symmetrical expansion of a spherical shell, 
    (with J. M. Hill), Q. J. Mech. Appl. Math. 49, pp 337-351 (1996)

15.   The depth of a steep evaporating grain boundary groove: Application of comparision theorems
      (with P.Broadbridge, P. Tritcher and Y. Karciga) Mathl. Comput. Modelling 25(10) , pp. 1-8 (1997)
16.   On the linearization of the Monge-Ampere equation  u_xxu_yy  -  u_xy² =  F(u,u_x/u_y)
      (with J. M. Hill), J. Nonlinear Math. Phys., 5(2), pp. 115-119 (1998)
17.   Ad-hoc solution methods in the context of Lie symmetries
      (with P. Broadbridge and J. M. Goard) in "22nd Internat. Colloga. On Group Theor. Methods in Physics" (1998)

18.    Transformations and equation reductions in finite elasticity. III A general class of first integral for plane strain deformations, (with J. M. Hill) Math. Mech. Solids 4 pp3-15 (1999)

19.   All solutions of standard symmetric linear partial differential equations have a classical Lie symmetry
      (with P. Broadbridge) J. Math. Anal. Appl. 234, pp109-122 (1999)

20.    A note on Ericksen’s problem for radially symmetric deformations in spherical polars with constant strain invariants, 
    (with J. M. Hill) Math. Mech. Solids 4, (1999)

21.    On the general structure of small on large problems for elastic deformations of Varga materials I: Plane strain deformations, (with J. M. Hill) J. Elasticity 54, pp. 193-212 (1999)

22.    On the general structure of small on large problems for elastic deformations of Varga materials II: Axially symmetric deformations, (with J. M. Hill) J. Elasticity 54, pp. 213-227 (1999)

23.    On the determining equations for the nonclassical reductions of the heat and Burgers' equation
    (with F. Hickling) ) J. Math. Anal. Appl. 270 No. 2, 582-589 (2002)

24.     A Darboux transformation for a class of linear parabolic partial differential equation
    (with F. Hickling) J. Phys. A: Math. Gen. 35, 389-399 (2002)

25.    An n^th order Darboux transformation for the one-dimensional time dependent Schrodinger equation
    (with F. Hickling) J. Phys. A: Math. Gen. 36, 1615-1621 (2003)

26.     Nonclassical symmetries of evolutionary partial differential equations and compatibility
    (with J. R. Beckham**) J. Math. Anal. Appl. 289, 55-65, (2004)

27.     Nonclassical contact symmetries and Charpit's method of compatibility,
    J. Non. Math. Phys., 12(3), 321-329, (2005)

* undergraduate student,     ** graduate student