Pages: 46

No chapter abstract available.

Pages: 28

In this introductory chapter, the fundamental principles of mathematical modeling and computer simulations are presented. First, a simple example of the free-falling object from a height using Mathematical Modeling is considered. To describe the trajectory of the object the questions “where” and “when” are posed. This leads to introduction of spatial coordinate and time. Next, the key word “gravitation” leads to a simple differential equation for the trajectory. This simple problem is considered following the steps for developing various types of mathematical models. The application of software tools Mathematica, MATLAB, and Python is described. Key topics such as model stability, dimensions, units, and scaling are briefly addressed at a beginner level. The chapter concludes with exercises and mini-projects to reinforce learning.

Pages: 16

This chapter provides an overview of key computational methods in applied mathematics. We explore numerical and symbolic computations for derivatives and integrals using software packages. Iterative direct and indirect methods are illustrated through Fibonacci numbers, and Newton's method for equation f(x) = 0 is discussed. The method of successive approximation is presented with examples of both convergent and divergent procedures. The Banach fixed-point theorem is introduced in both general and simplified forms, with applications to Cauchy's problem and equivalent integral equations. A symbolic expansion method is demonstrated through an example. The challenges of numerically solving certain equations are explained. The chapter concludes with exercises and mini-projects.

Pages: 92

No chapter abstract available.

Pages: 16

This chapter begins by the mechanical interpretation of derivatives: the first derivative of a path S(t) represents instantaneous velocity, and the second derivative represents acceleration. The relationship between integrals and energy is discussed, emphasizing the transition between continuous and discrete forms, where integrals are approximated by sums. The treatment of potential energy equations in phase space is explained. Polynomial interpolation is demonstrated through numerical examples. The chapter highlights the instability of differentiation and the stability of integration with discrete data, and it explains how to replace the integration of functions of several variables with summation. The chapter concludes with exercises and mini-projects.

Pages: 37

The theory of ordinary differential equations (ODEs) can be conditionally divided into the following three branches. The first branch deals with finding a solution to an ODE, emphasizing the classification of equations to apply appropriate solution techniques. The corresponding manipulations are applied to find a solution when the class is identified. Not all ODEs can be solved using elementary (well-known) functions. Frequently, one identifies a differential equation with a mathematical model. This stresses the exceptional role of differential equations in Mathematical Modeling. Computational tools such as Mathematica, MATLAB, or Python are essential for solving complex equations. The second branch involves qualitative analysis, addressing the existence, uniqueness, and stability of solutions. The third branch is related to applications in various fields, including mechanics, biology, economics, and chemistry, as well as numerical methods for solving ODEs that lack elementary or special function solutions. Key concepts include the principle of transition, exemplified by radioactive decay, and the logistic differential equation with its modifications. Topics such as stability analysis, bifurcation, and time delay are explored. The Taylor and Padé approximations are used. The harmonic oscillator with friction and exterior forces is investigated. The Lotka-Volterra model and the method of linearization are described. The chapter concludes with exercises and mini-projects.

Pages: 19

The method of least squares (MLS) is one of the prominent of stochastic analysis of data and processes. It can be applied to investigation of linear dependence between two sets X and Y of data and given by the regression equation. The correlation coefficient is measure of the linear dependence between X and Y. Fitting is represented as a not necessarily linear dependence between X and Y when the data approximation is found by MLS on the basis of elements. For instance, the basis {1, x, x2} yields the fitting function as a linear combination a + b x + c x2. The chapter includes method of Monte Carlo in its simple version of calculation the constant π and outline random walks. The chapter ends with exercises and mini-projects.

Pages: 18

The chapter is devoted to steady spatial processes reduced to one-dimensional problems. The heat conduction is modeled by the second-order differential equation https://www.w3.org/1998/Math/MathML" display="inline"> u ' ' x = f ( x ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781032684284/45759e50-f3f2-4cf5-885b-8a544c364d30/content/math_1.tif"/> on a segment with boundary conditions. One-dimensional Green's function is introduced and conceptualized as a heat source. The Dirac delta function is discussed, highlighting its connection to the Green's function. Additionally, the chapter includes animated code demonstrations.

Pages: 155

No chapter abstract available.

Pages: 29

This chapter covers the fundamentals of vector analysis, including scalar, vector, and mixed products, along with tensors. Rotation of a rigid body is represented in terms of vector analysis. Polar, cylindrical and spherical coordinates are described. After providing general definitions and properties, the chapter demonstrates the implementation of vector analysis in Mathematica. The theory of scalar and vector fields is introduced, with practical applications in both Mathematica and MATLAB. Key vector analysis operators such as gradient, divergence, curl, and Laplacian are detailed. Integral theorems of vector fields are outlined, and the chapter concludes with exercises and mini-projects.

Pages: 28

In this chapter, mathematical models for heat conduction are developed. One-dimensional problems were discussed in the previous Chapter 6. The heat equation, also known as the diffusion equation, serves as a model for various phenomena, including diffusion and electrostatics. This reflects a common scenario in mathematical modeling where a single equation applies across different scientific domains. The chapter details initial and boundary value problems for heat conduction, introducing Green's function and the method of separation of variables. Additionally, the finite difference and finite element methods are outlined.

Pages: 40

In this Chapter, we face with modeling formulas for the effective properties of composites within the framework of materials science. The attention is paid to conductivity (thermal, dielectric, etc.) and elasticity of deterministic and random composites. Simple observations reveal that some analytical formulas contradict each other because they are derived without considering homogenization principles and asymptotic rules. The prevalence of incorrect formulas and misleading methodologies used by engineers has compelled us to extend the discussion on mathematical modeling, utilizing the Manevitch - Andrianov asymptotic methodology. Maxwell's asymptotic approach is based on the assumption that inclusions are located far away from each other, and the interactions among inclusions are not considered. Maxwell's approach consists in two steps, solution to a single-inclusion problem and application of self-consistent method to estimate the effective constants up to the second order term in concentration O(f 2). The single-inclusion problem can be replaced by a finite set of inclusions. Then, the self-consistent method leads to the effective constants of dilute clusters. The method of analytical representative volume element valid for dispersed composites is briefly presented Densely packed inclusions are investigated by the structural asymptotic approximations.

Pages: 56

This section introduces machine learning and data analysis, which is currently one of the fast-paced research fields. The methods presented here are different from those presented before. They are summarized in a data-driven paradigm, where we do not create algorithms (step-by-step instructions) for producing outputs from inputs. Instead, we present some specifically prepared data to the program, which is responsible for finding patterns and generalizing them to other similar inputs. The chapter covers the description of supervised, unsupervised learning, and data preparation for machine learning. A simple example of classification problem and statistics are presented. Reading, cleaning and scaling data; dimensionality reduction by PCA; Selected models of supervised and unsupervised learning; regression and neural networks are discussed. The chapter concludes with an appendix being basic introduction to Python programming language.