1. Engineering Analysis – 804 441 Computational Fluid Dynamics – 804 416
Faculty Name
Prof. A. A. Saati

2. OR Computational Fluid Dynamics/Mechanics (CFD/CFM)
Engineering Analysis
OR Computational Fluid Dynamics/Mechanics (CFD/CFM)

4. Part 2 - Classification of ODE and PDE Equations
Classification of ODE Equations
Classification of PDE Equations

5. 1. Classification of ODE Equations
1.1 Classifying a differential equations as ordinary or partial:
Examples:
is an example of an ordinary differential equation
is an example of a partial differential equation

6. 1.1 Classifying a differential equations as ordinary or partial:
Examples:
An example of an ordinary differential equation
Assumption: Ball is a lumped system.
Number of Independent variables: One (t)
Hot Water
Spherical Ball

7. Examples: An example of partial differential equation
Assumption: Ball is not a lumped system.
Number of Independent variables: Four (r,θ,φ,t)
Hot Water
Spherical Ball

8. 1.2 Classifying a differential equation by order:
The order of an ordinary differential equation is the order of the highest-order derivatives present in the equation.
Examples:
the order is 1.
the order is 3.

9. 1.3 Classifying an ordinary differential
equation as linear or non- linear:
ODE is linear if:
That no products of the dependent variable and/or its derivatives are present.
No transcendental functions (sine, cosine, ey , arcsin, log, etc.) of the dependent variable and/or its derivatives are present.

10. 1.3 Classifying an ordinary differential
equation as linear or non- linear:
Examples: b
Linear
Nonlinear c

11. 2 - Classification of Partial Differential Equations (PDE)
Partial Differential Equations (PDEs).
What is a PDE?
Examples of Important PDEs.
Classification of PDEs.

12. 2.1 Introductory Remarks
A partial different ion equation (PDE) Is an equation that involves an unknown function and its partial derivatives

13. 2.1 Introductory Remarks
Notation

14. Partial differential Equations (PDE) is linear if:
That no products of the dependent variable and/or its derivatives are present.
No transcendental functions (sine, cosine, ey , arcsin, log, etc.) of the dependent variable and/or its derivatives are present.

15. Linear and Nonlinear PDEs

16. Representing the Solution of a PDE (Two Independent Variables)
Three main ways to represent the solution
T=5.2 t1
T=3.5 x1
2-D Plot
Different curves are used for different values of one of the independent variable
3-D Plot
Three dimensional plot of the function T(x,t)
The axis represent the independent variables.
The value of the function is displayed at grid points

17. Example 2-D Plot Different curves are used for different values of one of the independent variable
Different curve is used for each value of t
Heat Equation
ice
ice
Temperature
Temperature at different x at t=0 x
Thin metal rod insulated everywhere except at the edges. At t =0 the rod is placed in ice
Position x
Temperature at different x at t=h

18. Examples of PDEs
PDEs are used to model many systems in many different fields of science and engineering.
Important Examples:
Laplace Equation
Heat Equation
Wave Equation

19. Laplace Equation
Used to describe the steady state distribution of heat in a body.
Also used to describe the steady state distribution of electrical charge in a body.

20. Heat Equation
The function u(x,y,z,t) is used to represent the temperature at time t in a physical body at a point with coordinates (x,y,z)
 is the thermal diffusivity. It is sufficient to consider the case  = 1.

21. Simpler Heat Equation x
T(x,t) is used to represent the temperature at time t at the point x of the thin rod.

22. Wave Equation
The function u(x,y,z,t) is used to represent the displacement at time t of a particle whose position at rest is (x,y,z) .
The constant c represents the propagation speed of the wave.

24. Classification of PDE’s
Linear Second order PDEs are important sets of equations that are used to model many systems in many different fields of science and engineering.
Classification is important because:
Each category relates to specific engineering problems.
Different approaches are used to solve these categories.

25. Classification of PDE’s To classify the second-order PDE, Consider
the following equation
Where A,B,C,D,E,F and G are function of the independent variables x & y and dependent variable

26. Second-Order PDEs (contd.)
Assume that is a solution of the differential equation
The solution describes a surface in space on which curves may be drawn.
These curves are known as the characteristic curves and it is a patch various solutions of the differential equation.

27. Second-Order PDEs(contd.) Note:
The 2nd order derivatives along the characteristic curves are indeterminate and may be discontinuous
No discontinuity in the first derivatives is allowed.
The differentials of and represent changes from location to across the characteristics

28. Second-Order PDEs(contd.) Note:
The above 3 equations can be solved by using Cramer’s rule

29. Second-Order PDEs(contd.) Note:
Setting the denominator equal to zero
Yields to equation

30. Second-Order PDEs(contd.) Note:
The Classification of 2nd Order Linear PDE’s
Can be Classified based on as follow:
Elliptic if
Parabolic if
Hyperbolic if

31. Linear Second Order PDE Examples (Classification)

32. Linear Second Order PDE Examples (Classification)

33. Mixed Equations

34. Model Equations
The selected model PDEs which will be used in the next chapter are as follows
Laplace’s equation
Poisson’s equation
Unsteady heat conduction equation

35. Model Equations (contd.)
The y-component of the Navier-Stokes equation
The wave equation
The Burgers equation

36. Initial and Boundary Conditions
Initial Conditions:
An initial conditions is a requirement for dependent variable is specified at some initial stat
Boundary Condition:
A boundary conditions is a requirement for dependent variable or its derivative must satisfy on the boundary of the domain of PDE.

37. Initial and Boundary Conditions (contd.)
Types of Boundary Condition:
The Dirichlet boundary condition. If the dependent variable along the boundary is satisfy.
The Neumann boundary condition. If the normal gradient of the dependent variable along the boundary is satisfy
The Robin boundary condition. If the imposed boundary condition is a linear combination of types 1 & 2
The mixed boundary condition. If the boundary condition along a certain portion of the boundary is type 1, and on anther portion of the boundary is type 2

38. Boundary Conditions for PDEs
To uniquely specify a solution to the PDE, a set of boundary conditions are needed.
Both regular and irregular boundaries are possible. t
region of
interest x 1

39. The Solution Methods for PDEs
Analytic solutions are possible for simple and special (idealized) cases only.
To make use of the nature of the equations, different methods are used to solve different classes of PDEs.
The methods discussed here are based on the finite difference technique.

41. Example
Consider transient conduction in two-dimensions. Assume that along rectangular bar has been heated to a temperature distribution of
Initial condition
This is call an initial condition, such that
for t = 0
Boundary conditions
Now, place the bar in an environment in which the lower and right sides are in contact with a convicting fluid of and constant coefficient of h,
while the left side is insulated (adiabatic)
And the upper side is kept at a constant temperature

42. The corresponding boundary conditions are:

43. Remarks and Definitions
In order to solve a PDE by numerical methods the partial derivatives in the equation are approximated by finite difference relation
The resulting approximate equation, which represents the original PDE, is called a finite difference equation (FDE)
Example: Consider a two-dimensional rectangular domain. We wish to solve a PDE within this domain subject to imposed initial and boundary conditions.

44. Remarks and Definitions (contd.)
The rectangular domain is divided into equal increments in the x and y directions
Note that increment in the x direction do not need to be equal to the increments in the y direction.

45. Remarks and Definitions (contd.)
The location of mesh points, grid points, or nodes is designated by i an the x direction and by j in the y direction.

46. Remarks and Definitions

47. Remarks and Definitions

48. Remarks and Definitions

49. The End OF PART 2