Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania
Telephone: 610-758-4116. Fax: 610-758-3767.
Ph.D: Department of Mathematics, The Ohio State University. 1994 - 1999
Associate Professor: Department of Mathematics, Lehigh University.
14 East Packer Avenue, Bethlehem, Pennsylvania 18015 USA. 2008 - present.
Assistant Professor: Department of Mathematics, Lehigh University.
14 East Packer Avenue, Bethlehem, Pennsylvania 18015 USA. 2002 - 2008.
Dunham Jackson Assistant Professor: School of Mathematics, University of Minnesota.
Minneapolis, Minnesota 55455 USA. 1999 - 2002.
Existence and stability of traveling wave solutions of nonlinear integral differential equations
arising from synaptically coupled neuronal networks: By using rigorous mathematical analysis, I have accomplished
the existence and stability of traveling wave fronts of nonlinear scalar integral differential equations and
the existence and stability of traveling pulse solutions of nonlinear singularly perturbed systems of
integral differential equations arising from synaptically coupled neuronal networks.
I introduced speed index functions and stability index functions to accomplish the existence and stability of the traveling wave solutions.
Moreover, I have investigated the influence of various neurobiological mechanisms on the wave speeds of the traveling wave solutions.
Furthermore, I established bifurcations of standing wave solutions of both nonlinear singularly perturbed systems of integral differential equations
and nonlinear singularly perturbed systems of reaction diffusion equations.
Nonlinear systems of fluid dynamics equations: I have coupled together existing ideas, methods, results and new ideas to
generate a very different method to accomplish the existence and decay estimates with sharp rates of the global smooth solution
of a very general n-dimensional nonlinear system of fluid dynamical equations, including n-dimensional incompressible Navier-Stokes equations
and n-dimensional magnetohydrodynamics equations as examples. Moreover, I have established optimal stability estimates and exact limits
for the global smooth solution of the nonlinear system of fluid dynamical equations. I made use of an appropriate assumption on the nonlinear function.
Representations of bounded traveling wave solutions of five kinds of nonlinear evolution equations
(including nonlinear reaction diffusion equations, nonlinear dissipative dispersive wave equations, nonlinear dispersive wave equations,
nonlinear hyperbolic equations, and highly nonlinear, strongly coupled systems of differential equations):
By using some nonlinear transformations, by reducing higher order ordinary differential equations
to lower order ordinary differential equations, and by using the method of undetermined coefficients,
I have solved many second order nonlinear ordinary differential equations for particular solutions.
I applied them to find the bounded explicit traveling wave solutions of many nonlinear evolution equations.
PUBLICATIONS OTHER PEOPLE CITED OFTEN
Linghai Zhang, Decay of solutions of generalized Benjamin-Bona-Mahony equations.
Acta Mathematica Sinica, New Series. 10(1994), 428-438.
Linghai Zhang, Decay estimates for the solutions of some nonlinear evolution equations.
Journal of Differential Equations, 116(1995), 31-58.
Linghai Zhang, Sharp rate of decay of solutions to 2-dimensional Navier-Stokes equations.
Communications in Partial Differential Equations,
Linghai Zhang, Decay of solution of generalized Benjamin-Bona-Mahony-Burgers equations in n-space dimensions.
Nonlinear Analysis, Theory, Methods and Applications, 25(1995), 1343-1369.
Linghai Zhang, On stability of traveling wave solutions in synaptically coupled neuronal networks.
Differential and Integral Equations, 16(2003), 513-536.
Linghai Zhang, Existence, uniqueness and exponential stability of traveling wave solutions of
some integral differential equations arising from neuronal networks.
Journal of Differential Equations, 197(2004), 162-196.
Linghai Zhang, How do synaptic coupling and spatial temporal delay influence traveling waves
in nonlinear nonlocal neuronal networks?
SIAM Journal on Applied Dynamical Systems, 6(2007), 597-644.
more papers are coming.
I have taught both undergraduate and graduate classes at Lehigh University: including
Mathematics 21: Calculus One
Mathematics 22: Calculus Two
Mathematics 23: Calculus Three
Mathematics 205: Differential Equations and Linear Algebra
Mathematics 320: Ordinary Differential Equations
Mathematics 405: Partial Differential Equations
and many other classes.
Melissa Anne Stoner received her Ph.D in May 2011 (her research interest is in mathematical neuroscience).
Alan Dyson and Jie Yuan are working on their Ph.D degrees (both research interests are in mathematical neuroscience and dynamical systems).
The Lord is my shepherd, I shall not be in want.
He makes me lie down in green pastures,
he leads me beside quiet waters, he restores my soul.
He guides me in paths of righteousness for his name's sake.
Even though I walk through the valley of the shadow of death,
I will fear no evil, for you are with me;
your rod and your staff, they comfort me.
You prepare a table before me in the presence of my enemies.
You anoint my head with oil; my cup overflows.
Surely goodness and love will follow me all the days of my life,
and I will dwell in the house of the Lord forever.