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Published byCarolina Ridgeway Modified about 1 year ago

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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

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

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

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

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

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

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

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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.

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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, ……….

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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.

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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.

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Direction fields and solution curves

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

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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

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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)

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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)

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

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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)!!!

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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

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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

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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

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