Intro to Matlab
Intro to Matlab MATH 410
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This 2 page Class Notes was uploaded by Miss Hillary Grady on Thursday October 29, 2015. The Class Notes belongs to MATH 410 at College of William and Mary taught by Junping Shi in Fall. Since its upload, it has received 11 views. For similar materials see /class/231143/math-410-college-of-william-and-mary in Mathematics (M) at College of William and Mary.
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Date Created: 10/29/15
Math 410510 Notes 3 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 see Notes 1 for details on this model dP P BP2 ikP177 77 POP 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 8 7 27 A39 U We can use our table of dimensions above to check that these new variables are now dimensionless Step 3 Now use the chain rule to derive a new differential equation 1P 7 dP du 13 Adu B inu E E i E A 3 B712 1 uz Thus The term kP 1 7 becomes kAu lt1 7 and the term the equation simpli es to 713132 simpli es to A2 P2 A 7 Buz 1u2 u N du BikA 17 d3 ult Dividing by B gives du 7 kA 1 u U2 ds B 7quot NA 1 u Noting that the combinations of the parameters that occur above leads us to introduce two new dimensionless parameters 7 kA N 04E7 5 du 1 u U2 70m 7 d3 3 1u2 At the meantime7 the initial condition P0 P0 becomes 140 PoA through the change of variable u PA Thus if we introduce another new parameter 39y POA7 then the initial condition becomes The equation then becomes 140 W Note that the equation has two dimensionless variables 3 u and three dimensionless parameters a B V which are combinations of the original parameters This simpli ed form of the equation has reduced the number of parameters from 5 to 37 which makes the analysis of the equation simpler Exercises 1 Consider the equation dP P BP2 a1 PGNEZ3 Use the following change of variables Q P in ace in a 7 N s 7 7 7 s 7 7 to get nondimensionalized equations 2 Consider a population model 1 d7kz 17 7M 350 zo where k M7 N7 mo are positive parameters a In the following table7 ll in the dimensions of all parameters in terms of the dimensions of variables Variable Dimension Parameter Dimension t 739 m M N 0 b Use the change of variable yskt Derive the new equation including the initial condition in the new variables y and s
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