1. Chapter 7 Differential Equation Problems
Solving Applied Mathematical Problems with MATLAB
CRC/Taylor & Francis Press
Chinese version by Tsinghua University Press
PPT by Wenbin Dong and Jun Peng, Northeastern University, PRC
Proofread by Dingyu Xue & YangQuan Chen
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Slide 1 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

2. Chapter 7 MATLAB Solutions to Differential Equation Problems
Analytical Solutions of Differential Equations
Numerical Solutions to Ordinary Differential Equations
Numerical Solutions to Special Ordinary Differential Equations
Solving Boundary Value Problems
Introduction to Partial Differential Equations
Solving Ordinary Differential Equations using Simulink Blocks
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3. 7.1 Analytical Solutions of Differential Equations
Mathematical descriptions
Analytical solution methods
Applications of Laplace transforms
Analytical solutions to linear state-space equations
Analytical solutions to special nonlinear differential equations
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4. 7.1.1 Mathematical descriptions
A general description of linear ordinary differential equations with constant coefficients:
where are constants.
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5. For zero initial value problems,
Correspondingly, the Laplace transform is:
If the characteristic roots si are distinct roots, the analytical solutions become:
where Ci are constants to be determined and
g(t) is the special solution with respect to the input u(t) .
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6. 7.1.2 Analytical solution methods
Simple form, where the independent variable is assume to be t
With independent variable of x
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7. Example 7-1 Assume that the input signal is defined by
find the general solution to the ODE
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8. Assume that the following conditions are given
Results
Assume that the following conditions are given
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9. 星期四, , 22:08:34
Slide 9 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

10. the analytically solution to the equation can be obtained
Let
the analytically solution to the equation can be obtained
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11. The final solution is given by
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Slide 11 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

12. Example 7-2 For input signal , solve with 星期四, 2008-4- 24, 22:08:34
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13. Results
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14. Example 7-3
Find the analytical solution to the differential equation sets
Direct solution
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15. 7.1.3 Applications of Laplace Transforms
Transfer function:
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16. If U(s) is given as a rational function
Then, the output signal can be expressed, after rearrangement of terms, by the following s-domain function
With
If U(s) is given as a rational function
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17. Example 7-4
Given the input signal u(t)=e-5tcos(2t+1)+5 , assuming all zero initial conditions, solve analytically the differential equation
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18. step 1:
step 2:
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19. Analytical output signal in time domain:
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20. Using dsolve() function call
Step1:
Step2:
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21. The analytical solution is the same
Step3
The analytical solution is the same
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22. 7.1.4 Analytical solutions to linear state-space equations
Assume that the linear state-space model of a given system is
where A,B,C,D are constant matrices, The analytical solutions for such an equation is
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23. Example 7-5
For the input signal u(t) = 2 + 2e-3t sin 2t, find the analytical solutions to the system given by
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24. Solution by direct integration
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Slide 24 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

25. 7.1.5 Analytical solutions to special nonlinear differential equations
Very few nonlinear differential equations can be analytically solved by dsolve() function
An example for the analytical solution to a special nonlinear differential equation is demonstrated.
Other examples which cannot be solved are also demonstrated.
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26. Example 7-6 Solve nonlinear differential equation
Change the form of the original ODE:
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27. Example 7-7
Try to find the analytical solutions to the well-known Van der Pol equation.
The following command tested
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28. 7.2 Numerical Solutions to Ordinary Differential Equations
Overview of numerical solution algorithms
Fixed-step Runge-Kutta algorithm and its MATLAB implementations
Numerical solution to first-order differential equation sets
Transforms to standard ordinary differential equations
Validation of numerical solutions to ODEs
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Slide 28 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

29. 7.2.1 Overview of numerical solution algorithms
Explicit first-order differential equation
Standard form
where
State vector
Nonlinear functions
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Slide 29 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

30. 7.2.1.1 Numerical error and step-size problem
Euler algorithm:
Consider IVP. Given x(t0), the initial state vector at time instant .
The derivative in the left hand can be approximated by a simple Euler formula
The solution can be approximated by
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31. At time instant , the state vector is denoted by
So, suppose at any time instant tk ,the state vector is xk then, at the , Euler algorithm is given by
h is called the step size.
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Slide 31 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

32. Two reasons that we cannot reduce the step size h unlimitedly:
Slow down in computation
Increase the cumulative errors
Three considerations in numerically solving ODEs:
Suitable step-size selection
Improved algorithms
Variable-step-size algorithms
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Slide 32 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

33. 7.2.2 Fixed-step Runge-Kutta algorithm and its MATLAB implementations
The fourth-order fixed step size Runge-Kutta algorithm:
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Slide 33 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

34. The state vector for the next step:
where h is the step size
The state vector for the next step:
MATLAB syntax
Loop structure can be used in the function
See next slide the MATLAB implementations
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35. 星期四, , 22:08:34
Slide 35 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

36. 7.2.3 Numerical solution to first-order differential equation sets
Runge-Kutta-Felhberg algorithm
The current step is hk. Define 6 variables ki:
State vector for the next step:
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Slide 36 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

37. MATLAB functions for solving ordinary differential equations
Define an error vector:
MATLAB functions for solving ordinary differential equations
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Slide 37 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

38. Prepare the ODE's right hand side function (RHSF) under study as:
Modify the option parameters:
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Slide 38 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

39. Solution Procedures Write mathematically the equation
Write MATLAB code for the equation
M-function
Anonymous function
Inline function, not recommended
Solve equation
Draw results
Verification of results
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Slide 39 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

40. Example 7-8 Solve the following Lorenz model parameters
The initial values are
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41. Describing the equation
M-function
Anonymous function
Inline function (not recommended)
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Slide 41 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

42. MATLAB command for solutions
Phase space trajectory
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Slide 42 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

43. Solving ODEs in MATLAB with additional arguments
Why to introduce additional arguments
In the previous example, if parameters
changes, no need to modify function
additional arguments can be used
New function format
New calling command
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Slide 43 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

44. Example 7-9
Prepare a MATLAB function with parameters for Lorenz equation. Then use the new function to find the numerical solutions for another set of parameters
with additional parameters
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45. Solution procedure in MATLAB
Phase space trajectory
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Slide 45 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

46. Choose new parameters New calling command 星期四, 2008-4- 24, 22:08:34
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Slide 46 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

47. 7.2.4 Transforms to standard ordinary differential equations
Manipulating a single high-order ODE
General high order ODE:
with zero initial conditions
select a set of state variables as
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48. Then, the original high order ODE can be written as a set of first order ODEs:
with initial values
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49. Example 7-10 Van der Pol equation With initial Selecting states
New ODE
MATLAB code
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50. MATLAB solutions A counter-example 星期四, 2008-4- 24, 22:08:34
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Slide 50 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

51. A counter-example For the following parameters
Suggestions: DO NOT RUN the following
Stiff equation algorithms should be used
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52. Manipulating multiple high-order ODEs
State choosing method is not unique
Suggestions: choose the state variables:
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53. The new state equation is then
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Slide 53 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

54. Example 7-11 Apollo satellite trajectory (x, y) is governed by
where and
Initial states
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55. Define a set of state variables: After conversion, the new ODE
Where
And
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56. MATLAB description 星期四, 2008-4- 24, 22:08:34
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57. Solutions Verification 星期四, 2008-4- 24, 22:08:34
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58. Step size plot Comments Should not rely too much on defaults
Better set precision control to
Step size should be as small as
Solutions should be verified
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59. Example 7-12 Fixed step size Runge-Kutta algorithm where
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60. Using a fixed step size 0.01:
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Slide 60 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

61. Example 7-13 Solve the differential equation sets
Select the state variables:
and
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62. Substitute into the second eqn to find
The first order ODE can be constructed
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63. With solve function One also finds 星期四, 2008-4- 24, 22:08:34
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64. Transformation of differential matrix equations
The Lagrange equation
Where matrices
column vectors
Selecting state vector
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65. the state-space equation can be rewritten as
Where
Thus the matrix equation can be solved with the ode45() or ode15s() functions
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66. Example 7-14 Consider the inverted pendulum model expressed as
where q = [a,q1,q2]T, and a is the position of the cart, and q1, q2 are the angles of the two bars. The matrices are given by
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67. The parameters 星期四, 2008-4- 24, 22:08:34
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68. MATLAB description 星期四, 2008-4- 24, 22:08:34
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69. Solving the inverted pendulum problem with the following MATLAB code
Original system unstable
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Slide 69 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

70. 7.2.5 Validation of numerical solutions to ODEs
It has been shown through examples that if the control parameters are not properly chosen, the solutions may not be correct. Thus the numerical solutions should be validated.
The most effective control parameter is the RelTol property. The default value for it is 10-3, which is too large. It can be set respectively to 10-6 or even 10-8.
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71. 7.3 Numerical Solutions to Special Ordinary Differential Equations
Solutions of stiff differential equations
Solutions of implicit differential equations
Solutions to differential algebraic equations
Solutions to delay differential equations
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72. 7.3.1 Solutions of stiff differential equations
In the Van der Pol equation,
In many differential equations, the solutions to some states changes very rapidly while others may change very slowly.
Stiff equations.
An alternative function, ode15s(), can be used instead and this function has exactly the same syntax as that of the ode45() function.
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73. Example 7-15
Find the numerical solutions to Van der Pol equation, when
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74. Example 7-16 Solve the ODE Regarded as stiff in classical courses
The analytical solutions
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75. Numerical solutions to the problem
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76. Using a fixed step size:
For better results, one should reduce the step size, however slow
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77. Example 7-17 Numerically solve the ODE Initial condition Solutions
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78. Using variable step size:
Using ode15s() instead of ode45():
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79. 7.3.2 Solutions of implicit differential equations
The so-called implicit differential equations are those who cannot be converted into the first-order explicit equations.
The implicit equations can still be solved by the use of explicit equations.
Or use of solver ode15i
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80. Example 7-18 Given x1(0)=x2(0)=0 , numerically solve
By denoting x=[x1,x2]T, re-write the original ODE as follows:
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81. where
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82. Example 7-19 Given x(0)=[1,0,0,1]T, numerically solve
Define the state vector:
Re-write the original implicit ODE:
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83. Solve the Algebraic equation in the model
Describe the equation in MATLAB
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84. continued from overleaf
The derivatives can be evaluated
Solution process:
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85. Use of ode15i():
It is a new ODE command in MATLAB 7.0 for solving implicit ODEs directly.
General description of implicit ODEs:
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86. Example 7-20
Using implicit ODE solution method, for initial state x(0)=[1,0,0,1]T
State variables:
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87. The differential equation can be rewritten as
Recall the form
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88. Solutions
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89. 7.3.3 Solutions to differential algebraic equations
A more general form of the differential equations:
is singular
Differential algebraic equation
Cannot use
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90. Example 7-21
Given initial conditions, x1(0)=0.8, x2(0)=x3(0)=0.1, Numerically solve the following DAE:
Using matrix form description for this DAE:
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91. Solving the equation 星期四, 2008-4- 24, 22:08:34
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92. Alternative solution method
We can also transform this DAE to the normal ODE. From the algebraic constraint:
by substitution, we get
An anonymous function can be written to describe the differential equations
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93. Solving again
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94. To use ode15i() command 星期四, 2008-4- 24, 22:08:34
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95. Example 7-22 Solve when 星期四, 2008-4- 24, 22:08:34
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96. 7.3.4 Solutions to delay differential equations
General form of the DDEs
where is the time delay for the state x(t)
Implicit Runge-Kutta algorithm dde23()
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97. Example 7-23 Numerically solve this DDE: Initial states
Select the state variables:
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98. M-function description
New standard form
M-function description
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99. MATLAB solutions 星期四, 2008-4- 24, 22:08:34
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100. Example 7-24 Neutral-type DDE Cannot be solved with dde23() function.
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101. 7.4 Solving Boundary Value Problems
Solutions of two-point boundary value problems
Solutions of general boundary value problems
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102. The above are the conditions at two boundaries a and b.
General mathematical description of boundary value problem for second-order differential equations:
interval of interest: [a, b]. Study the solution when the following boundary conditions are specified:
The above are the conditions at two boundaries a and b.
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Slide 102 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

103. 7.4.1 Solutions of two-point boundary value problems
Standard form
Problem can be converted as
Newton iteration:
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104. where Solving this 星期四, 2008-4- 24, 22:08:34
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Slide 104 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

105. where it is required to explicitly solve Prepare the MATLAB function:
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Slide 105 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

106. Example 7-25
Solve the following boundary value problem to the nonlinear differential equations
Derive the partial derivatives:
We get
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Slide 106 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

107. Analytical solution is Verification:
Solution process:
Analytical solution is
Verification:
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Slide 107 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

108. 7.4.2 Solutions of general boundary value problems
bvp5c() function for general BVPs
Parameter initialization
MATLAB descriptions to differential equations and boundary value problems
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Slide 108 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

109. Example 7-26
Solve again the boundary value problem in Example 7.25 with the bvp5c() solver.
Select states
The new state equation
MATLAB description
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Slide 109 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

110. Rewrite the boundary conditions
The problem can be solved with
Pay attention to the last command
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Slide 110 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

111. If the boundary value problem is It can be rewritten as
The solution can be found by
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Slide 111 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

112. Example 7-27 For a differential equation With Four conditions
find the parameters and solve the BVP
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Slide 112 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

113. The equations can be written as
Let
The equations can be written as
MATLAB description
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Slide 113 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

114. The problem can be solved with
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Slide 114 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

115. 7.5 Introduction to Partial Differential Equations
Solving a set of PDEs
Mathematical description to the two-dimensional PDEs
The GUI for the PDE Toolbox --- an introduction
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Slide 115 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

116. 7.5.1 Solving a set of PDEs
The pdepe() command can be used to solve the following class of PDEs
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Slide 116 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

117. Boundary conditions can be specified as
Initial conditions
Simple MATLAB solutions of IC
Solutions of PDE
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118. Example 7-28 Solve the following PDE 星期四, 2008-4- 24, 22:08:34
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Slide 118 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

119. The function Initial conditions Boundary conditions
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Slide 119 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

120. Change the original PDE as the following:
Clearly, here , and also
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Slide 120 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

121. The M-function describing the PDE
Specify the boundary conditions
left bounds
right bounds
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Slide 121 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

122. Prepare the MATLAB function for BC:
Prepare the MATLAB function for IC:
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Slide 122 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

123. Solution process: 星期四, 2008-4- 24, 22:08:34
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Slide 123 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

124. 7.5.2 Mathematical description to the two-dimensional PDEs
Four kinds of PDEs
Elliptic
Parabolic
Hyperbolic
Eigenvalue type
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Slide 124 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

125. Elliptic PDEs General description of elliptic PDE: Gradient
where
Gradient
Divergence:
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Slide 125 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

126. If c is a constant, the above can be simplified as
Elliptic PDEs:
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Slide 126 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

127. Parabolic PDEs The general form of a parabolic PDE
If c is a constant, the above equation can be simplified as
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Slide 127 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

128. Hyperbolic PDEs The general form of a hyperbolic PDE
If c is a constant, the above equation can be simplified
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Slide 128 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

129. Eigenvalue PDEs An eigenvalue PDE problem is defined as
If c is a constant, the above equation can be simplified as
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Slide 129 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

130. 7.5.3 The GUI for the PDE Toolbox --- an introduction
PDE Toolbox --- an overview
Start the PDE Toolbox GUI
Type pdetool in MATLAB
Four elements in this GUI
Menu system
Toolbar
Set formula
Solution regions
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Slide 130 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

131. Drawing and defining the region in PDE solution
the button labeled as can be used to define the solution region.
the button labeled as D, can be used to made triangular mesh within the solution region.
Boundary conditions for PDEs
Dirichlet BC:
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Slide 131 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

132. Examples of PDE solution via PDE Toolbox
Neumann BC:
Expanded form:
where is the partial differential of u,w,r,t . The vector normal to x.
Examples of PDE solution via PDE Toolbox
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133. Example 7-29 Solving the following hyperbolic PDE:
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Slide 133 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

134. Animation of the time varying solutions
Solving PDEs when parameters are not constants
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Slide 134 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

135. Example 7-30
Assume that the partial differential equations are described as
which is an elliptic PDE where
Solve this PDE with u=0 on the boundary.
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Slide 135 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

136. 7.6 Solving Ordinary Differential Equations using Simulink Blocks
An introduction to Simulink
Simulink --- relevant blocks
Using Simulink for modeling and simulation of ODEs
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Slide 136 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

137. 7.6.1 An introduction to Simulink
1990 – first appearance of Simulink, with an old name of SimuLAB, renamed Simulink
Simulink: two meanings
simulation (simu) and model/block
interconnection/link (link)
Use odegroup to open self-defined model block library.
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Slide 137 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

138. 7.6.2 Simulink --- relevant blocks
Commonly used blocks:
Input and output port blocks (In1, Out1)
Clock block: generates time variable t
Commonly used input blocks:
Sine
Step
Constant
Integrator block (Int): just to integrate the dynamic system
Transport delay blocks (Transport Delay)
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Slide 138 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

139. Mathematical operation blocks Mathematical function blocks
Gain blocks:
Gain
Sliding Gain
Matrix Gain
Mathematical operation blocks
+, -, *, ÷
Mathematical function blocks
Sin, exp
Signal vectorization blocks:
Mux
DeMux
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Slide 139 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

140. 7.6.3 Using Simulink for modeling and simulation of ODEs
Build the corresponding Simulink model for ODE.
Use sim( ) command to solve the Simulink block model.
Get the ODE numerical solution.
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Slide 140 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

141. Example 7-31
Solve Lorenz equation: with b=8/3, r=10, s=28, and initial conditions x1(0)= x2(0)= 0, x3(0)=10-10, Use Simulink to construct this model and obtain the simulation result.
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Slide 141 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

142. Establish the block diagram Create model c7mlor1b.mdl Solve the system
Compare with ode45 results
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Slide 142 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

143. Example 7-32 Solve the following DDE using Simulink.
Create model c7mdde2.mdl
Solution process:
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Slide 143 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

144. The first order state space model:
Assign the new states
The first order state space model:
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Slide 144 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

145. Example 7-33
Where t1=0.15, t2=0.5
Use Simulink to build this DDE system.
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Slide 145 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

146. Enter the known matrices:
Create the model c7mdde3.mdl
Solution process:
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147. Chapter summary Functions 星期四, 2008-4- 24, 22:08:34
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148. 星期四, , 22:08:34
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149. For general nonlinear ODE, it is not possible to obtain its analytical solution. So, must rely on numerical solutions.
This chapter introduced the concept of numerical solution of ODEs starting with the simplest Euler algorithm. Then, for general numerical solutions, we introduced the basic idea and the related concept of step size. The MATLAB ODE solver ode45( ) has been introduced with several demo examples.
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Slide 149 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

150. In initial value problem of ODEs can be solved easily when they are in the first-order explicit equations. If not, we introduced the conversion method using the concept of state variables. In particular, we introduced how to convert the high order ODEs into a set of first-order ODEs.
Matrix differential equations are discussed, one may first convert the matrix differential equations into first-order explicit equations, then the equations can be solved.
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Slide 150 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

151. When the ODE solution process is very slow, it is possible that the ODE is stiff and we must use stiff ODE solvers such as ode15s() for a reliable and fast solution. Additionally, we introduced other types of ODEs such as differential algebraic equations (DAEs), implicit ODEs, and delay differential equations (DDEs). The relevant MATLAB solution techniques for these special types of ODEs were introduced with examples.
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Slide 151 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

152. For the boundary value problem (BVP) for second order ODEs, we can use the universal solver bvp5c() with examples.
For partial differential equations (PDEs), we can use the ready commands provided in MATLAB PDE Toolbox. For 2D PDEs, we can use MATLAB PDE Toolbox via the GUI. This PDE toolbox is also useful for high dimensional PDEs.
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Slide 152 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008

153. Simulink is an important dynamic system simulation platform within MATLAB. We can use the block diagram connections to construct any given dynamic system model using the existing Simulink block library. In this book, we focused more on Simulink basic knowledge and more on ODEs and DDEs.
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Slide 153 (of 153) Dingyü Xue and YangQuan Chen, Solving Applied Mathematical Problems with MATLAB, CRC Press, 2008