Ordinary Differential Equations Swaroop Nandan Bora swaroop@iitg.ernet.in Department of Mathematics Indian Institute of Technology Guwahati Guwahati-781039
Modelling a situation We study a model, a sort of idealized world that contains things we do not see come across in everyday life. We study straight lines, rectangles, circles, spheres. NOT a burger or a chair or a hill or a human being. If one works in a practical area of mathematics then there will be two conflicting criteria which makes a good model. On the one hand, the model should be accurate enough to be useful. On the other hand, it should be simple and elegant enough to generate realistic and interesting mathematical problem.
Modelling a situation It is tempting, as a mathematician, to attach far more importance to the second criterion: mathematical interest and elegance rather to the first: accuracy.
Mathematicians Approach In particular, if mathematicians work on difficult practical problems they do not do so in isolation from the rest of mathematics. Rather, they bring to the problems several tools mathematical tricks, rules of thumb, theorems known to be useful and so on. They do not know in advance which of these tools they will use, but they hope that after they have thought hard about a problem they will realize what is needed to solve it. If they are lucky, they can simply apply their existing expertise straightforwardly. More often, they will have to adapt it to some extent.
Practical Problems: Applicable mathematics LET s now move to the practical side of mathematics in various stages. In doing so, we need to restrict ourselves since applications are also as vast as the subject itself.
Tea/Coffee Figure : Hot Coffee
Tea/Coffee Figure : Cold Coffee
Falling Body Figure : Falling Body
Pendulum Figure : Motion of a pendulum
Impulsive Force Figure : Impulsive Force
River Figure : River flow with current
River Figure : Quiet River flow
Sloshing Figure : Sloshing
Building Construction Figure : Building Construction
Flow through porous media Figure : Aquifer
Ocean wave mechanics Figure : An ocean wave
Ocean Engineering Figure : Rectangular platform in ocean
Ocean Engineering Figure : Offshore Oil drilling platform
Figure : An aeroplane in its flight
General Information Many of the general laws of nature in physics, chemistry, biology and astronomy find their most natural expression in differential equations. Applications are mainly in the areas of mathematics itself, engineering, economics and many other fields of applied sciences. Why is it so??
Differential Equations We know that if y = f(x) is a given function, then its derivative dy can be interpreted as the rate of change of y with respect to x. dx In many natural processes, the variables involved and their rates of changes are connected to one another by means of the basic scientific principles that govern the process. When this connection is expressed in mathematical symbols, the result is quite often a differential equation. Let us consider some examples we already know.
Example 1 According to Newton s second law of motion, the acceleration a of a body of mass m is proportional to the total force F acting on it, with 1/m as the constant of proportionality, so that a = F/m or ma = F. (1) Suppose, for instance, that a body of mass m falls freely under the action of gravity alone, then the only force acting on it is mg. If y is the distance down to the body from some fixed height, then its velocity v = dy is the rate of change of position and its acceleration dt a = dv dt = d2 y is the rate of change of velocity. dt2 With this notation, (22) becomes m d2 y dt 2 = mg, or, d 2 y = g. (2) dt2
Example 1 If we change the situation by assuming that there is an air resistance proportional to the velocity, then the total force acting on the body is mg k(dy/dt). (22) becomes m d2 y = mg kdy dt2 dt. (3) Equations (??) and (??) are the differential equations that express the essential attributes of the physical processes under consideration. They are respectively called undamped and damped motion of the body.
Example 2 Newton s Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature (i.e. the temperature of its surroundings). Newton s Law makes a statement about an instantaneous rate of change of the temperature. When we translate this verbal statement into mathematical symbols, we arrive at a differential equation. The solution to this equation will then be a function that tracks the complete record of the temperature over time.
Example 2 If T is the temperature of an object at time t and S is the temperature of its surroundings, then this law formulates into dt = k(t S), (4) dt where k is a constant of proportionality. If T 0 is the initial temperature, the temperature of the object at any time t is given by T(t) = S +(T 0 S)e kt. (5)
Example 3 Consider a pendulum of length l whose bob has a mass m Then the equation of motion (undamped case) is given by Is this the equation we usually know? d 2 θ dt 2 + g sinθ = 0. l Or the equation we know is different from this? The accepted form is the linearized version d 2 θ dt 2 + g l θ = 0.
Boundary and Initial Conditions Boundary conditions are conditions prescribed on the boundary Boundary may be boundary with respect to any of the independent variables Initial conditions are conditions prescribed at one point only These conditions are in terms of some form of the dependent variable at some specific value of the independent variable The main component of this type of problems is what is called Governing Equation
Boundary and Initial Conditions (Contd.) With respect to ODEs we can have only boundary conditions or only initial conditions, not both for the same problem They are called boundary value problems or initial value problems. However, with respect to PDEs We can have both boundary conditions and initial conditions for the same problem This type of problems are called Initial Boundary Value Problems (IBVP)
BVPs and IVPs A boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. In other words, a solution to a BVP is a solution to the differential equation which also satisfies the boundary conditions. To be useful in applications, a BVP should be well-posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. An initial value problem (IVP) consists of a differential equation and a set of conditions to be satisfied at the initial value of the independent variable (for ODE) or at that of one of the independent variables (for PDE).
BVPs and IVPs A more mathematical way to picture the difference between a BVP and an IVP is an IVP has all of the conditions specified at the same value of the independent variable in the equation (and that value is at the lower value of the boundary of the domain, thus the term initial value), while a BVP has conditions specified at the extremes of the independent variable. For example for a second-order differential equation if the independent variable is time over the domain [0,1], an IVP would specify a value of y(t) and y (t) at time t = 0, to be precise, the initial conditions will be something like y(0) = α,y (0) = β. On the other hand a BVP would specify values for y(t) (or its derivatives) at both t = 0 and t = 1, to be precise, the boundary conditions will be something like y(0) = α 1,y(1) = β 1 or y (0) = α 2,y (1) = β 2.
IBVPs If the problem is dependent on both space and time (meaning the governing equation is a PDE) then instead of specifying the value of the problem at a given point for all time only, data could be given at a given time for all space also. This type of problems is known as initial boundary value problems (IBVP). Prime examples are the problems involving the wave equations and the transient heat conduction equations.
Types of conditions The type of boundary conditions that will be considered for a BVP will depend on the dimension of the object under consideration. For example for the heat conduction in a thin rod, the boundaries will be the two end points of the rod while for a thin rectangular plate, the boundary will consist of the four edges that bound the plate
Types of conditions If the boundary conditions are prescribed in terms of some values of the dependent variable (solution of the BVP) on the boundary, then these conditions are called Dirichlet Conditions and the corresponding BVPs are called Dirichlet boundary value problems If the boundary conditions are prescribed in terms of some values of the normal derivatives on the boundary, then these conditions are called Neumann conditions and the corresponding BVPs are called Neumann boundary value problems. Neumann conditions are also known as flux conditions. If there is no flux across the boundary, then the flux conditions become insulation conditions (for heat conduction problems).
Types of conditions If the boundary conditions for a specific problem contain both types, then these conditions are called mixed or Robin conditions and the corresponding BVPs are called Robin boundary value problems. For heat conduction problems, Robin conditions are also known as radiation conditions. A typical problem in heat conduction may have a combination of Dirichlet, flux/insulation and radiation boundary conditions.
Continuity conditions Sometimes there may be virtual boundaries. Say For a fluid problem, the fluid is of two layers. Then at the boundary of the layers, called interface, there exist some conditions known as continuity conditions They usually imply continuity of pressure and velocity along the boundary
Idealizations Given to us: a real life problem Can we solve the problem exactly with the given conditions? Perhaps not. Under this circumstances we need to idealize the situation, that is, we want to ignore some of the given situations/properties in order to obtain a feasible solution. What do we do?
Idealizations Idealization can take place in three ways Property Governing Equation Boundary Conditions We need to idealize the situation, that is, we want to ignore some of the given situations/properties in order to obtain a feasible solution.
Moving ahead Recall Newton s Second Law of Motion In terms of differential equation, it is F = ma and the solution can be obtained as Given two initial conditions the arbitrary constants can be eliminated d 2 y dt 2 = g y = gt2 2 +c 1t+c 2
Moving ahead Does the equation represent the real situation? Perhaps not. What was ignored?? The air resistance the particle encounters while falling down If air resistance is taken into account, m d2 y dt 2 = mg kdy dt A new term is appended due to the air resistance The former equation is an idealized version of the latter one.
Moving ahead Now consider the pendulum equation What was ignored?? The air resistance the bob encounters while moving from one end to the other Now what we will have a damped version of the equation.
Remarks There are many functions/polynomials which are solutions of some specific ODEs. They are called special functions, such as Bessel function, Legendre polynomial, Hermite polynomial, Laguerre polynomial etc. More interestingly, they are part of Mathematical Physics rather than only of Mathematics.