Introduction to Mathematical Modeling and Computer Simulations
portes grátis
Introduction to Mathematical Modeling and Computer Simulations
Kycia, Radoslaw Antoni; Rylko, Natalia; Mityushev, Vladimir; Nawalaniec, Wojciech
Taylor & Francis Ltd
12/2024
330
Dura
9781032661513
15 a 20 dias
Descrição não disponível.
I. General Principles and Methods. 1. Principles of Mathematical Modeling. 1.1. How to develop a mathematical model. 1.2. Types of models. 1.3. Stability of models. 1.4. Dimension, units, and scaling. 2. Numerical and symbolic computations. 2.1. Numerical and symbolic computations of derivatives and integrals. 2.2. Iterative methods. 2.3. Newton's method. 2.4. Method of successive approximations. 2.5. Banach Fixed Point Theorem. 2.6. Why is it difficult to numerically solve some equations? Exercises and mini-projects. II. Basic Applications. 3. Application of calculus to classic mechanics. 3.1. Mechanical meaning of the derivative. 3.2. Integral and energy. 3.3. Potential energy. 3.4. Interpolation. 3.5. Integration of discrete functions. Exercises and mini-projects. 4. Ordinary differential equations and their applications. 4.1. Principle of transition for ODE. 4.2. Radioactive decay. 4.3. Logistic differential equation and its modifications. 4.4. Time delay. 4.5. Approximate solution to differential equations. 4.6. Harmonic oscillation. 4.7. Lotka-Volterra model. 4.8. Linearization. Exercises and mini-projects. 5. Stochastic models. 5.1. Method of least squares. 5.2. Fitting. 5.3. Method of Monte Carlo. 5.4. Random walk. Exercises and mini-projects. 6. One-dimensional stationary problems. 6.1. 1D geometry. 6.2. Second order equations. 6.3. 1D Green's function. 6.4. Green's function as a source. 6.5. The ?-function. III. Advanced Applications. 7. Vector analysis. 7.1. Euclidean space R3. 7.2. Scalar, vector and mixed products. 7.3. Rotation of bodies. 7.4. Scalar, vector, and mixed product in Mathematica. 7.5. Tensors. 7.6. Scalar and vector fields. 7.7. Integral theorems. Exercises and mini-projects. 8. Heat equations. 8.1. Heat conduction equations. 8.2. Initial and boundary value problems. 8.3. Green's function for the 1D heat equation. 8.4. Fourier series. 8.5. Separation of variables. 8.6. Discrete approximations of PDE. 8.7. Universality in Mathematical Modeling Table. Exercises and mini-projects. 9. Asymptotic methods in composites. 9.1. Principle of Asymptotology. 9.2. Effective properties of composites. 9.3. Principles of homogenization theory. 9.4. Maxwell's approach. 9.5. Mathematical modeling and effective properties of composites. 9.6. Strategy of investigations. 9.7. Densely packed balls. Exercises and mini-projects. 10. Machine learning and data analysis. 10.1. Supervised, unsupervised learning, and regression. 10.2. Data storage. 10.3. A simple example of classification problem. 10.4. Reading, cleaning and scaling data. 10.5. Simple statistics. 10.6. Dimensionality reduction by PCA. 10.7. Selected models of supervised learning. 10.8. Selected models of unsupervised learning. 10.9. Regression. 10.10. Neural Networks. A. Introduction to Python.
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numeric and symbolic computations;calculus;stochastic models;vector analysis;asymptotic methods;Wojciech Nawalaniec;Natalia Rylko
I. General Principles and Methods. 1. Principles of Mathematical Modeling. 1.1. How to develop a mathematical model. 1.2. Types of models. 1.3. Stability of models. 1.4. Dimension, units, and scaling. 2. Numerical and symbolic computations. 2.1. Numerical and symbolic computations of derivatives and integrals. 2.2. Iterative methods. 2.3. Newton's method. 2.4. Method of successive approximations. 2.5. Banach Fixed Point Theorem. 2.6. Why is it difficult to numerically solve some equations? Exercises and mini-projects. II. Basic Applications. 3. Application of calculus to classic mechanics. 3.1. Mechanical meaning of the derivative. 3.2. Integral and energy. 3.3. Potential energy. 3.4. Interpolation. 3.5. Integration of discrete functions. Exercises and mini-projects. 4. Ordinary differential equations and their applications. 4.1. Principle of transition for ODE. 4.2. Radioactive decay. 4.3. Logistic differential equation and its modifications. 4.4. Time delay. 4.5. Approximate solution to differential equations. 4.6. Harmonic oscillation. 4.7. Lotka-Volterra model. 4.8. Linearization. Exercises and mini-projects. 5. Stochastic models. 5.1. Method of least squares. 5.2. Fitting. 5.3. Method of Monte Carlo. 5.4. Random walk. Exercises and mini-projects. 6. One-dimensional stationary problems. 6.1. 1D geometry. 6.2. Second order equations. 6.3. 1D Green's function. 6.4. Green's function as a source. 6.5. The ?-function. III. Advanced Applications. 7. Vector analysis. 7.1. Euclidean space R3. 7.2. Scalar, vector and mixed products. 7.3. Rotation of bodies. 7.4. Scalar, vector, and mixed product in Mathematica. 7.5. Tensors. 7.6. Scalar and vector fields. 7.7. Integral theorems. Exercises and mini-projects. 8. Heat equations. 8.1. Heat conduction equations. 8.2. Initial and boundary value problems. 8.3. Green's function for the 1D heat equation. 8.4. Fourier series. 8.5. Separation of variables. 8.6. Discrete approximations of PDE. 8.7. Universality in Mathematical Modeling Table. Exercises and mini-projects. 9. Asymptotic methods in composites. 9.1. Principle of Asymptotology. 9.2. Effective properties of composites. 9.3. Principles of homogenization theory. 9.4. Maxwell's approach. 9.5. Mathematical modeling and effective properties of composites. 9.6. Strategy of investigations. 9.7. Densely packed balls. Exercises and mini-projects. 10. Machine learning and data analysis. 10.1. Supervised, unsupervised learning, and regression. 10.2. Data storage. 10.3. A simple example of classification problem. 10.4. Reading, cleaning and scaling data. 10.5. Simple statistics. 10.6. Dimensionality reduction by PCA. 10.7. Selected models of supervised learning. 10.8. Selected models of unsupervised learning. 10.9. Regression. 10.10. Neural Networks. A. Introduction to Python.
Este título pertence ao(s) assunto(s) indicados(s). Para ver outros títulos clique no assunto desejado.
