Non-linear FO-Partial Differential Equations P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Emanation from Better Understanding of the laws of nature to systems…..

Examples of FO-PDEs Linear FO-PDE: Semi-linear FO-PDE: Quasi-linear FO-PDE: Nonlinear FO-PDE: where F is not linear in Properties of solutions of all 4 classes of equations are quite different, but there is an universal Method of Solution……

René Descartes Biography Academic, Philosopher, Mathematician, Scientist (1596–1650) Descartes added theology and medicine to his studies. Descartes believed that all truths were ultimately linked. He introduced Cartesian geometry.

Geometrical Interpretation of FO-PDEs Consider simple FO-PDE: General solution in D =2 f is an arbitrary C1 function. Consider another simple FO-PDE: General solution in D =2 g is an arbitrary C1 function.

Quasilinear Equations: Geometric Approach Consider the quasilinear partial differential equation in two independent variables, Think geometrically. Identify the solution u(x, y) with its graph, This is x surface in xyz-space defined by z = u(x, y). Geometric Interpretation: If f(x, y, u) =0 is a solution of this equation, then this function describes the solution surface, or integral surface. The initial data along the  curve generates a space curve .  must lie on the graph and its also called as the initial curve of the solution.

The Central Idea The objective of a Quasi-linear FO PDE is to build the remainder of the graph as a collection of additional space curves that “emanate from”

Application of Vector Calculus Using vector calculus that the normal to the integral surface f(x,y,u) is given by the gradient function. Now consider the vector of coefficients given in original PDE. The dot product of coefficient vector with the gradient is : This is the left hand side of the partial differential equation.

Quasilinear FO-PDE as Vector Equation f is normal to the surface of coefficient vector. And hence, coefficient vector is tangent to the surface. It can be seen as the graph of the solution u(x, y) is made up of integral curves (stream lines) of the vector field f . Moral: The graph of the solution to the PDE is constructed by finding the stream lines of f that pass through the initial curve. This is equivalent to solving a system of ODEs! Geometrically, coefficient vector defines a direction field, This is called as the characteristic field.

Space Curve & The Method of Characteristics Parametrize the initial space curve using intrinsic geometry  =(t). The tangent to the curve is then

The Space Curve in QL- FO-ODE Equate above equations to get the parametric form of the characteristic curves as

Characteristic & Compatibility Equations These are called as characteristic equations of given QL-FO-ODE. is called as compatibility equations of given QL-FO-ODE. with a(x, y,u)  0 characteristic equations give Compatibility condition becomes

How do we use these Characteristic & Compatibility Equations to solve quasilinear partial differential equations?

Example Find the solution to with initial space curve Characteristic equations are : or What is the shape of these characteristic curves ?

Solution Compatibility conditions along these curves are with

In Search of A General Solution with The goal is to find the general solution to the differential equation. Since u = f(x, y), the integration “constant” is not really a constant, but is constant with respect to x. It is in fact an arbitrary constant function. In fact, this can be viewed it as a function of C1, the constant of integration in the characteristic equation.

This is a general solution to the partial differential equation The General Solution Hence, let C2 = G(C1), G is an arbitrary function. Since The general solution of the given Partial differential equation can be written as This is a general solution to the partial differential equation

The Specific Solution The general solution: The side condition

The Final Solution

An Universal Method of Flying Fast Sky is the Limit

The Jet Plane Vampire The first to exceed a speed of 500 miles per hour. A total of 3,268 Vampires were built in 15 versions, including a twin-seat night fighter, trainer and a carrier-based aircraft designated Sea Vampire. DH108 was a newer version was built and released for test. Initially DH 108 behaved very nicely. As the speed was stepped up in was unsuspectingly drawn closer to an invisible wall in the sky. It was unknown to anyone. One evening the pilot hit this wall and the plane was disappered.

1940-50’s Flying Story Cruising at High Altitudes ?!?!?! Aircrafts were trying to approach high altitudes for a better fuel economy. This led to numerous crashes for unknown reasons. These included: The Mitsubishi Zero was infamous for a peculiar unknown problem, and several attempts to fix it only made the problem worse. The rapidly increasing forces on the various surfaces, which led to the aircraft becoming difficult to control to the point where many suffered from powered flight into terrain when the pilot was unable to overcome the force on the control stick. In the case of the Super-marine Spitfire, the wings suffered from low torsional stiffness.

More Stories The P-38 Lightning suffered from a particularly dangerous interaction of the airflow between the wings and tail surfaces in the dive that made it difficult to "pull out“. Flutter due to the formation of thin high pressure line on curved surfaces was another major problem, which led most famously to the breakup of de Havilland Swallow and death of its pilot, Geoffrey de Havilland, Jr.