Introduction:

What is the role of this class in science or in your field? It is one of powerful tools along with experiments, observations and derivations.

This course is continuation of MA580 (numerical analysis with focus on linear algebra) and is one of subjects of the qualifying exams for Ph.D candidates in Mathematics. The focuses of this course are various numerical methods and analysis for problems including: approximation and interpolation,  numerical differentiation and integration, numerical solution of initial value problems for ordinary differential equations,  fast Fourier transform, and numerical solution of non-linear equations of one or several variables.

This course is designed for students in applied mathematics, engineering, and other disciplines to learn basic techniques in numerical analysis and applications. While theoretical foundations will be described, emphasis will also be placed on algorithm design and implementation. We will also explore available software packages in this field.

Problem solving process:  A flow chart:

Real problems   --- Physical Laws/Other approaches <---> Mathematical/physical Models (Statistics, Integral Equations, optimization, ..., Differential equations) --> Solution techniques: Analytic solutions, Approximate solutions --> Interpret the solutions --> Applications (products, experiments, better models, predictions).

Solution Techniques:

Analytic solutions: If the solution(s) can be expressed in terms of elementary functions (sin(x), cos(x), e^x, log(x)),  series, or other known functions. But unfortunately, for many and many problems, it is extremely difficult, if not impossible, to get an analytic solution. Sometimes, the analytic solution may be too complicated and one would prefer to use an approximations.

Approximate Solutions:

Analytic approximate solution technique. Simplify the problem so that an analytic solution is obtainable; Or approximate solution by a simpler solution. For example, perturbation analysis; asymptotic expansions, truncated Taylor expansions or Fourier series etc. Usually do not use computers except symbolic software packages such as Maple, Mathematica etc.

Numerical Solution Techniques: Discretize the original differential equations in some form, then use computers to solve the discrete systems. The solution will be presented as: numerical solution (numbers);  or some simples function constructed from the computed numbers; or visualized by plots.

Technical terms:

Numerical Mathematics, Computational Mathematics. A branch of Mathematics/Applied Mathematics, the science that uses computers to solve mathematical problems.

 

Numerical Analysis, similar to computational mathematics but with the emphasis on theoretical analysis such as convergence and stability.

 

Scientific Computing,  similar to computational mathematics but with the emphasis on getting computational results. The justification of a valid method is either with rigorous theoretical proof or comparison with experiments or observed data or physical phenomenon.