With his Ph . D . thesis " " Randwertproblemmethoden zur Parameteridentifizierung in Systemen nichtlinearer Differentialgleichungen " " ( Boundary-value problem methods for parameter estimation in systems of nonlinear differential equations ) completed under the supervision of Jens Frehse and Roland Z . Bulirsch, he received a Ph . D . in applied mathematics from the University of Bonn in 1986.
32.
He is best known for his work in chemical engineering and hydrodynamics including the approximate methods for solving nonlinear differential equations of mass, heat, and momentum transfer; mathematical modeling of chemical reactor processes and catalytic distillation; heat, mass, and momentum transfer in turbulent flow; fluid dynamics in granular beds; surface convection ( Marangoni instability ), absorption, and molecular convection.
33.
When he was exposed there to classical human and animal data about learning, the philosophical paradoxes that were implicit in these data triggered an intellectual inquiry that led him to introduce, during his freshman year, the modern paradigm of using nonlinear differential equations with which to describe neural networks that model brain dynamics, as well as the basic equations that many scientists use for this purpose today ( see Research ).
34.
To find the minimums a variational method is used, resulting in a set of nonlinear differential equations, called " Brown's equations " after William Fuller Brown Jr . Although in principle these equations can be solved for the stable domain configurations " "'M " "'( " "'X " "'), in practice only the simplest examples can be solved.
35.
Where \ textstyle a : = \ frac { k _ f } { k _ i } is the ratio of post-yield \ textstyle k _ f to pre-yield ( elastic ) \ textstyle k _ i : = \ frac { F _ y } { u _ y } stiffness, \ textstyle F _ y is the yield force, \ textstyle u _ y the yield displacement, and \ textstyle z ( t ) a non-observable hysteretic parameter ( usually called the " hysteretic displacement " ) that obeys the following nonlinear differential equation with zero initial condition ( \ textstyle z ( 0 ) = 0 ), and that has dimensions of length:
36.
In 1986, Kruskal and Zabusky shared the Howard N . Potts Gold Medal from the Franklin Institute " for contributions to mathematical physics and early creative combinations of analysis and computation, but most especially for seminal work in the properties of solitons . " In awarding the 2006 Steele Prize to Gardner, Greene, Kruskal, and Miura, the American Mathematical Society stated that before their work " there was no general theory for the exact solution of any important class of nonlinear differential equations . " The AMS added, " In applications of mathematics, solitons and their descendants ( kinks, anti-kinks, instantons, and breathers ) have entered and changed such diverse fields as nonlinear optics, plasma physics, and ocean, atmospheric, and planetary sciences.
