Mathematical Model

Why modeling

*      Fundamental and quantitative way to understand and analyze complex systems and phenomena
*      Complement to Theory and Experiments, and often Integrate them
*      Becoming widespread in: Computational Physics, Chemistry, Mechanics, Materials, …, Biology

Mathematical Modeling?
Mathematical modeling seeks to gain an   understanding of science through the use   of mathematical models on HP computers.

Mathematical modeling involves teamwork


Complements, but does not replace, theory and experimentation in scientific research.

*      Is often used in place of experiments when experiments are too large, too expensive, too dangerous, or too time consuming.
Is a modern tool for scientific investigation











Mathematical Modeling Process


Real World Problem
Identify Real-World Problem:
n   Perform background research, focus on a workable problem.
n   Conduct investigations (Labs), if appropriate.
n   Learn the use of a computational tool: Matlab, Mathematica, Excel, Java.
Understand current activity and predict future behavior. 
Example: Find the sum of two arrays.
Working Model
Simplify à Working Model:                         
Identify and select factors to describe important aspects of Real World Problem; deter mine those factors that can be neglected.
n  State simplifying assumptions.
n  Determine governing principles, physical laws.
Identify model variables and inter-relationships
Principle: c=a+b

Mathematical Model
Represent à Mathematical Model: Express the Working Model in mathematical terms; write down mathematical equations whose solution describes the Working Model.
In general, the success of a mathematical model depends on how easy it is to use and how accurately it predicts.
c[0]=a[0]+b[0]
c[1]=a[1]+b[1]
.
.
.
c[n]=a[n]+b[n]
Computational Model
Translate à Computational Model:  Change Mathematical Model into a form suitable for computational solution.
*      Existence of unique solution
*      Choice of the numerical method
*      Choice of the algorithm
*      Software
Input:   array a[] i.e. {1,2,3,8,5}
            Array b[] i.e. {2,5,7,4,5}
Output: Resultant array c[]
Initialize: c[0:4]=0
Time stepping:
1.      Read array a[]
2.      Read array b[]
3.      If i>=0
a.       c[i]=a[i]+b[i]
b.      i++
c.       goto step 3
4.      print c.

Results/Conclusions
Simulate à Results/Conclusions: Run “Computational Model” to obtain Results; draw                       Conclusions.
n    Verify your computer program; use check cases; explore ranges of validity.   
n    Graphs, charts, and other visualization tools are useful in summarizing results and drawing conclusions.  

Real World Problem
Interpret Conclusions: Compare with Real World Problem behavior.
n    If model results do not “agree” with physical reality or experimental data, reexamine the Working Model (relax assumptions) and repeat modeling steps.
n    Often, the modeling process proceeds through several iterations until model is “acceptable”.






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