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Section 1.3: Differential Equations As Mathematical Models a Mathematical Model Is a Mathematical Description of a Physical Syst

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Section 1.3: Differential Equations As Mathematical Models a Mathematical Model Is a Mathematical Description of a Physical Syst Section 1.3: Differential Equations as Mathematical Models A mathematical model is a mathematical description of a physical system or phenomenon. Dynamical System: A system that changes or evolves in time t. • Consists of a set of time-dependent variables, called state variables • and rules (i.e. mathematical model) for determining the state of the system at any time in terms of a specified state at some time t0. The state of the system at time t is the value of the state variables • at t; the specified state at time t0 is the initial condition. Classified as either discrete-time systems or continuous-time systems, • dependent on whether the state variables are defined at discrete times or over a continuous time interval. For a continuous-time dynamical system, the mathematical model (or • rule) is a DE or system of DEs, and the specified state of the system at time t0 is the initial condition. This gives an IVP. The solution of the IVP is called the response of the system. • Static systems are systems that do not evolve in time. 1 Steps in Model Development: 1. Make a clear formulation of the real system or phenomenon to be mod- eled, identifying the variables that control the system. 2. Choose the variables to include in the model (i.e., specify the level of resolution). 3. Make reasonable assumptions about the system, including any applicable empirical laws. There are different levels of assumptions. 4. Formulate the model. This often consists of a set of ODEs and/or PDEs, as well as side conditions (e.g., initial conditions, boundary conditions). 5. Solve the set of derived equations. This can be difficult and most often requires numerical techniques. 6. Compare model predictions with reality. Make comparisons in terms of known behavior or trends, or common sense (qualititative compari- son). Whenever possible, make comparisons with measured data from experiments (quantitative comparison). 7. If good agreement, then use the model (until it is proven bad); If unsatisfactory agreement, then modify level of resolution or assump- tions. That is, go back to step 2 or 3. 2 Example: Falling Bodies and Air Resistance A body of weight w falls from rest under the influence of gravity and a retarding force resulting from air resistance. It is assumed that the force du to air resistance is proportional to the velocity of the body. Find an ODE for the velocity of the body at any time t. Then formulate an appropriate IVP. Recall: Newton's Second Law of Motion: F = m a where F = sum of forces acting on a body m = mass of the body a = acceleration of the body Let s = distance the body has fallen from its initial position 3 Example: Falling Bodies and Air Resistance (cont) From Calculus: ds Velocity: v(t) = dt dv d2s Acceleration: a(t) = = dt dt2 Forces acting on body: 1. Gravity: acts downward, that is, in the positive direction, and is equal to the weight of the body F1 = w = m g where g is the acceleration due to gravity 2. Air resistance: acts upward, that is, in the negative direction, and is assumed to be proportional to the velocity of the body F = k v 2 − where k > 0 is a constant 4 Example: Falling Bodies and Air Resistance (cont) Total force on the body: F = F + F = m g k v 1 2 − Therefore, Newton's Second Law of Motion (F = m a) gives: dv m g k v = m − dt dv k = = g v ) dt − m This is a first-order, linear ODE. IVP: dv k = g v, t > 0 dt − m v(0) = 0 5 Example: Mixtures (Problem 10, page 28) Suppose a large mixing tank initially holds 300 gallons of water into which 50 pounds of salt have been dissolved. Another brine solution is pumped into the tank at a rate of 3 gal/min and, when the solution is well stirred, it is pumped out at a slower rate of 2 gal/min. If the concentration of the solution entering the tank is 2 lbs/gal, determine a differential equation for the amount A(t) of salt in the tank at time t. 6.
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