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Understanding the Importance of Numerical Methods in Mathematical Modeling and Problem Solving

Numerical methods are essential tools for solving complex mathematical problems that cannot be addressed analytically. They apply computational algorithms to obtain approximate solutions for various applications, including heat flow on wires, air flow around airplane wings, and responses to medication. By simplifying calculations through creating conceptual models and using heuristics, these methods enable predictions and facilitate analysis of equilibrium equations through continuous and discrete experiments. This exploration discusses various numerical techniques, their effectiveness, and considerations such as convergence and error analysis.

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Understanding the Importance of Numerical Methods in Mathematical Modeling and Problem Solving

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Presentation Transcript

  1. Why do we need Numerical Methods?

  2. Conceptual model

  3. Examples: Heat flow on a wire Air flow around a plane’s wing Response to medication Conceptual model

  4. Examples: Heat flow on a wire Air flow around a plane’s wing Response to medication Conceptual model Equilibrium Equations Heuristics Simplifications Continuous Discrete Experiments, Statistics, Prediction Mathematical Model

  5. Examples: Heat flow on a wire Air flow around a plane’s wing Response to medication Conceptual model Equilibrium Equations Heuristics Simplifications Continuous Discrete Experiments, Statistics, Predictions Mathematical Model other nonlinear ODE linear

  6. Examples: Heat flow on a wire Air flow around a plane’s wing Response to medication Conceptual model Simplifications Continuous discrete Experiments, statistics, prediction Mathematical Model other nonlinear ODE linear Ax=b Find A-1 x=A-1b ?????

  7. Examples: Heat flow on a wire Air flow around a plane’s wing Response to medication Conceptual model Simplifications Continuous discrete Experiments, statistics, prediction Mathematical Model other nonlinear ODE linear f(x) =0 Newton’s Method ???????? Ax=b Find A-1 x=A-1b ?????

  8. Examples: Heat flow on a wire Air flow around a plane’s wing Response to medication Conceptual model Simplifications Continuous discrete Experiments, statistics, prediction Mathematical Model other nonlinear ODE linear f(x) =0 Newton’s Method ???????? Euler’s Method ???????? Ax=b Find A-1 x=A-1b ?????

  9. Examples: Heat flow on a wire Air flow around a plane’s wing Response to medication Conceptual model Simplifications Continuous discrete Experiments, statistics, prediction Optimal control, PDEs, DAEs,… Mathematical Model other nonlinear ODE linear f(x) =0 Newton’s Method ???????? Euler’s Method ???????? Ax=b Find A-1 x=A-1b ?????

  10. Examples: Heat flow on a wire Air flow around a plane’s wing Response to medication Conceptual model Simplifications Continuous discrete Experiments, statistics, prediction Optimal control PDEs, DAEs,… Mathematical Model other nonlinear ODE linear f(x) =0 Newton’s Method ???????? Euler’s Method ???????? Ax=b Find A-1 x=A-1b ????? Numerical Methods

  11. Examples: Heat flow on a wire Air flow around a plane’s wing Response to medication Conceptual model Simplifications Continuous discrete Experiments, statistics, prediction Optimal control, PDEs, DAEs,… Mathematical Model other nonlinear ODE linear f(x) =0 Newton’s Method ???????? Euler’s Method ???????? Ax=b Find A-1 x=A-1b ????? Numerical Methods Computed Solution

  12. Numerical Methods Computed Solution • Algorithms • Programming • Convergence • Error • Analysis • Stability • What can go wrong? • Which algorithm works best • on which type of problem?

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