Technique of nondimensionalization Aim: –To remove physical dimensions –To reduce the number of parameters –To balance or distinguish different terms in the equation –To choose proper scale for different variables Method: –Set a scale for each variable –Plug into the equation & balance different terms –Determine the scales

Example 1 For Malthus model Set By the chain rule Plug into the equation

Example 1 Simplify the equation Choose Plug back, we get the dimensionless equation No parameter !!

Example 2 For logistic model Set By the chain rule Plug into the equation

Example 2 Simplify the equation Choose Plug back, we get the dimensionless equation No parameter in the equation !!!

Example 3 For insect outbreak model Set By chain rule: Plug into the equation

Example 3 Simplify the equation Scaling 1: Choose Plug back, we get

Example 3 Scaling 2: Choose Plug back, we get The scaling is NOT unique. All are correct. Different ones are good for different parameter regime.

Analytical and numerical solutions We obtain first ODE from different applications –Mixture problem –Population models Malthus model Logistic model Logistic model with harvest –Point-mass motion –Maximum profit –Rocket, ……….

Analytic & Numerical Solutions of 1st Order ODEs General form of first order ODE: –t: independent variable (time, in dimensionless form) –y=y(t): state variable (population, displacement, in dimensionless form) –F=F(t,y): a function of two variables Solution: y=y(t) satisfies the equation –Existence & uniqueness ??? –Geometric view point: at various points (t,y) of the two-dimensional coordinate plane, the value of F(t,y) determines a slope m=y’(t)=F(t,y)!! –A solution of this differential equation is a differential function with graph having slope y’(t) at each point through which the graph passes.

Analytic & Numerical Solutions of 1st Order ODEs –Solution curve: the graph of a solution of a differential equation –A solution curve of a differential equation is a curve in the (t,y)- plane whose tangent at each point (t,y) has slope m=F(t,y). Graphical method for constructing approximate solution: –Direction field (or slope field): through each of a representative collection of points (t,y), we draw a short line segment having slope m=F(t,y). –Sketch a solution curve that threads its way through the direction field in such a way that the curve is tangent to each of the short line segments that it intersects. –Isoclines: An isoclines of the differential equation y’(t)=F(t,y) is a curve of the form F(t,y)=c (c is a constant) on which the slope y’(t) is constant.

Direction fields and solution curves

Direction fields by software In Mathematica: –plotvectorfield In Maple: –DEplot In Matlab: –dfield: drawing the direction field

Solutions of ODE Infinitely many different solutions!!! Solution structure of simple equation –If y 1 (t) is a solution, then y 1 (t)+c is also a solution for any constant c ! –If y 1 (t) and y 2 (t) are two solutions, then there exists a constant c such that y 2 (t)=y 1 (t)+c. –If y 1 (t) is a specific solution, then the general solution is (or any solution can be expressed as) y 1 (t)+c

Solutions of ODE Infinitely many different solutions!!! Solution structure of linear homogeneous problem –If y 1 (t) is a solution, then c y 1 (t) is also a solution for any constant c ! –If y 1 (t) and y 2 (t) are two nonzero solutions, then c 1 y 1 (t) + c 2 y 2 (t) is also a solution (superposition) and there exists a constant c such that y 2 (t)=c y 1 (t). –If y 1 (t) is a specific solution, then the general solution is c y 1 (t)

Solutions of ODE Solution structure of linear problem –If y 1 (t) is a solution of the homogeneous equation, y 2 (t) is also a specific solution. Then y 2 (t)+c y 1 (t) is also a solution for any constant c ! –If y 1 (t) and y 2 (t) are two solutions, y 1 (t)-y 2 (t) is a solution of the homogeneous equation. –If y 1 (t) and y 2 (t) are specific solutions of the homogeneous equation and itself respectively, then any solution can be expressed as y 2 (t) +c y 1 (t)

Solutions of IVP Initial value problem (IVP): Solution: –Existence –Uniqueness Examples –Example 1: Solution: There exists a unique solution

Solutions of IVP –Example 2 Solution: separable form General solution Different cases: –b=0: these is at least one solution y=y(t)=0 –b>0 (e.g. b=1): these is no solution!! (F(t,y) is not continuous near (0, b>0)!!!

Solutions of IVP Theorem: Suppose that the real-valued function F(t,y) is continuous on some rectangle in the (t,y)-plane containing the point (t 0,b) in its interior. Then the above initial value problem has at least one solution defined on some open interval J containing the point t 0. If, in addition, the partial derivative is continuous on that rectangle, then the solution is unique on some (perhaps smaller) open interval J 0 containing the point t=t 0. Proof: Omitted

Solutions of IVP Example 1: –Condition: F and are continuous –Conclusion: There exists a unique solution for any initial data (t 0,b) Example 2: –Condition: F is continuous for y>=0 and is continuous for y>0 –Conclusion: There exists a unique solution for any initial data (t 0,b>0) For b=0, e.g. y(0)=0, there are two solutions

Classification –Linear ODE, i.e. the function F is linear in y –Otherwise, it is nonlinear –Autonomous ODE, i.e. F is independent of t Some cases which can be solved analytically