Intro to Mathematical Biology
Intro to Mathematical Biology MATH 345
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This 1 page Class Notes was uploaded by Miss Hillary Grady on Thursday October 29, 2015. The Class Notes belongs to MATH 345 at College of William and Mary taught by Junping Shi in Fall. Since its upload, it has received 10 views. For similar materials see /class/231144/math-345-college-of-william-and-mary in Mathematics (M) at College of William and Mary.
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Date Created: 10/29/15
Math 345 Notes Nondimensionalization J unping Shi The form of a solution of a differential equation can depend critically on the units one chooses for the various quantities involved Frequently these choices can lead to substantial problems when numerical approximation techniques such as Euler s method are applied These dif culties can be controlled or avoided by proper scaling We describe a technique that changes variables so that the new variables are dimensionless This technique will lead to a simple form of the equation with fewer parameters It makes clear that the parameters can interact in the equation and a simpler combined parameter can suffice for more than one parameter We illustrate this technique which is called Nondimensionalization with an example Example Consider the following model of an outbreak of the spruce budworm dP P BPZ 7kP177 77 P0P 1 dt lt N A2P2 H 0 H We give a step by step approach to nondimensionalize this initial value problem Step 1 List all of the variables parameters and their dimensions For the dimensions we use 739 for time and p for population in number of worms Variable Dimension Parameter Dimension 25 739 k 1 739 P p N p A p B p 739 P0 p Step 2 Take each variable and create a new variable by dividing by the combination of parameters that has the same dimension in order to create a dimensionless variable Not that there is not always a unique way to do that so some experimentation may be necessary Here we create P Bt 7 3 7 A7 A We can use our table of dimensions above to check that these new variables are now dimensionless LL Step 3 Now use the chain rule to derive a new differential equation dPidP du 13 du B du 7777iA Bi dt du 13 dt 13 A 13 2 2 The term kP 1 7 becomes kAu lt1 7 and the term simpli es to 15722 Thus the equation simpli es to du Au Buz B7 kA 1 7 7 7 ds 7quot lt N gt 1 u Dividing by B gives Lug 1L L ds B NA 1u2
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