Continuous Mathematical Models 2-Mathematical Modeling and Simulation-Lecture Slides, Slides of Mathematical Modeling and Simulation

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Continuous Mathematical Models

Mathematical Modeling

& Simulation

Continuous Mathematical Models

A very large portion of science and engineering is composed of mathematical modeling that represent physical systems and processes. These models are constructed using a few basic principals.

Mostly these contain derivatives to represent changes in system elements.

Coupling many systems means that we are connecting these differential equations.

Coupled Models

For coupled models we always need additional information for those variables and their relationships. This will produce a set of simultaneous or coupled equations as opposed to one equation.

Most of the science and engineering models involve coupled equations and their simulation.

Real systems have differential equations (in general less than the number of variables in the system).

Then possible relationships are provided by empirical algebraic equations, and arbitrary functions.

How to Classify a Model

To provide a framework, it is good to consider different ways of classifying models based on differential equations.

In next slides we first learn how to classify these models using properties of differential equations.

Number of independent variables: Partial

Differential Equations

When the unknown function depends on several independent variables, partial derivatives appear in the equation. In this case the equation is said to be a partial differential equation (PDE). Examples:

( , ) ( , ) (waveequation)

( , ) ( , ) (heat equation)

2

2 2

2 2

2 2

2 2

t

u x t x

a u x t

t

u x t x

u x t

 

Number of unknown functions: Systems

of Differential Equations

• Another classification of differential equations depends on the number of unknown functions that are involved.
• If there is a single unknown function to be found, then one equation is sufficient. If there are two or more unknown functions, then a system of equations is required.
• For example, predator-prey equations have the form

where u(t) and v(t) are the respective populations of prey and predator species. The constants a,c, ,  depend on the particular species being studied.

• In some areas of study, there may be thousands of equations.

dv dt cv uv

du dt au uv

 

  /

/

Linearity:

A differential equation based model is linear if none of the dependent term has products of dependent variables and/or any of its derivatives.

The model is non-linear if it has any dependent variable or its derivative term raised to power more than one, or any product term has dependent variable and its derivative as product.

Linearity:

Linear & Nonlinear Differential Equations

• An ordinary differential equation

is linear if F is linear in the variables

• Thus the general linear ODE has :
• Example: Determine whether the equations below are linear or nonlinear. ddty tddty t u uu t u u u t

y y y e y t y y t xx yy xx yy

y ( 4 ) 1 ( 5 ) sin ( 6 ) sin( ) cos

( 1 ) 3 0 ( 2 ) 3 2 0 ( 3 ) 3 2 0 (^44222)

2       

       

F  t^ , y , y ,^ y , y ,, y (^ n )^  0

y , y , y , y , , y ( n )

a 0 (^) ( t ) y ( n^ )  a 1 ( t ) y ( n ^1 )  an ( t ) y  g ( t )

Solutions:

Solutions to Differential Equations

A solution (t) to an ordinary differential equation

satisfies the equation. It means:

Example: Verify the following solutions of the ODE

y  y  0 ; y 1 ( t )sin t , y 2 ( t )cos t , y 3 ( t )  2 sin t

y (^ n^ )( t )  f  t^ , y , y , y ^ ,, y ( n ^1 )

 (^ n^ )( t )  f  t ,,,^ ,, ( n ^1 )

Conditions in the Model:

• Differential equations always have conditions at constant value of the independent variable.
• These are called initial conditions when we define them at a starting point.
• When we have conditions specified at two or more

points, they are termed as boundary conditions.

• We may have mixed conditions for a system i.e. one set of conditions for one type of model equations and the other set for the second type.

Example: Identify major features of the following mathematical model:

ddt y 3 yddty 2 5 dydt 2 y 5 t 8 cos 5 t 2 3

3 with    ^  y ( 0 ) 0 ., y ( 0 ) 1. 0 , y ( 0 )   0. 1

Solution: Dependent and independent variable: y and t,

Order: 3; Linearity: It is a non-linear equation as it has a product term: 2

2 dt yd y

Homogeneity: It is non-homogeneous equation with a force term ( 5t + 8cost ). Conditions: Initial conditions are given. Coefficients: There are constant coefficients (notice 3y with second derivative term is causing non-linearity). Driving term type: It has analytical term as opposed to a tabular form. Model Equation Type: It is a single ordinary differential equation based model. Docsity.com

Models Based On n’th Order ODEs

• Ordinary differential equations emerge when we study rates of change of dependent variables and have one independent variable.
• These are formed as a result of balance of forces; balance of energy in a given differential volume; balance of flows in a volume or in a surface or it can appear as a result of balancing momentums.
• We first from a differential surface or a volume under observation in a system.

Models Based On n’th Order ODEs

This may be decomposed into a set of first – order differential equations for digital computer simulations. :

Let us define a series of new variables

1 1 0

1

1 dt a y t

dy t

a

dt

d y t

f t a

dt

d y t

a n

n n n

n

n ^     





























 1

1

2

1

( ) (^ )

( ) ( )

( ) ( )

n

n n (^) dt g t d y t

dt g t dy t

g t y t



Models Based On n’th Order ODEs

After substituting these new variables, we convert the Equation (into a first order differential equation of the following form:

Dividing the above equation by an gives:

The above equation and Eqs. 1.1 constitute the decomposition of higher order differential equation (Eq. 1.1) into a set of first order differential equations.

( ) 1 ( ) 1 2 ( ) 0 1 ( ) f t a g t a g t a g dt

dg t a (^) n n   n  n  

( ) ( ) 1 ( ) 12 ( ) 0 g 1 ( t )

a

g t a

a

g t a

a

f t a

dt

dg t

n n

n n

n   n   