When tackling time series forecasting, many problems don't require jumping straight to deep learning models. Structures like trends, seasonality, autocorrelation, and volatility clustering can be cleanly expressed by traditional models — and they're interpretable, diagnosable, and make excellent baselines. This article places ARIMA/SARIMA (stationarity and seasonality), VAR (multivariate dynamics), GARCH (volatility modeling), Exponential Smoothing and Prophet (trend decomposition), and Kalman Filtering (state-space perspective) within a unified framework: what structure each model captures, what its core assumptions are, how to interpret parameters, and when each approach is more stable and worth trying first on real data.

When you press the power button on your computer, what hidden hardware orchestration lies behind the glowing screen? Why does your computer sometimes run like lightning, and other times crawl like a slideshow? From the basics of Bits and Bytes, to multi-core CPU architecture, three-level memory caches, SSD flash lifespan, and GPU parallel computing — this article takes you deep into every corner of computer hardware. We won't just tell you "what it is," but more importantly, explain "why it's designed this way." Through everyday analogies, real-world cases, and troubleshooting experience, you'll truly understand how computers work and never feel lost when facing hardware issues again.

Our differential equations journey is nearing its end, but mathematics itself is endless. This chapter introduces several active research frontiers — Neural ODEs, delay differential equations, stochastic differential equations, and fractional differential equations — which are profoundly changing our understanding of dynamical systems. Finally, we review the core content of the entire series and provide readers with a complete learning roadmap.

Differential equations are not a pure mathematical game — they are the language for understanding the physical world. From celestial motion to circuit response, from spring vibration to chemical reactions, almost all dynamical system behaviors can be described by differential equations. In this chapter, we explore the core applications of ordinary differential equations in physics and engineering, building a bridge from mathematics to practice.

When you drive a car, you constantly adjust the steering wheel to keep the vehicle centered in the lane; an air conditioner automatically regulates cooling power based on temperature feedback; a rocket precisely controls thrust to maintain orbit. These seemingly different systems share a common mathematical foundation — control theory. In this chapter, we explore how differential equations describe and design control systems, from classical PID controllers to modern state-space methods, seeing how mathematics helps us tame complex dynamical systems.

Why do lynx and snowshoe hare populations exhibit remarkable periodic fluctuations? Why does introducing a new species sometimes lead to ecological disaster? Why can some species coexist while others inevitably face competitive exclusion? The answers to these questions lie in differential equations. In this chapter, we explore the mathematical theory of population dynamics, from simple single-species growth to complex multi-species interactions, seeing how mathematics reveals the deep laws of ecosystems.

In early 2020, a novel coronavirus swept across the globe, and mathematical models became key tools for understanding the pandemic and formulating policies. When epidemiologists predicted that "if no measures are taken, the number of infections will grow exponentially," they were relying on differential equations. In this chapter, we will start from the simplest SIR model and gradually build a mathematical framework for understanding infectious disease transmission, validating these models' predictive power with real data.

When physical quantities change not only with time but also with space, we enter the realm of partial differential equations (PDEs). Heat conducting through a metal rod, waves spreading across water, electromagnetic fields propagating through space — these phenomena all require PDEs to describe. This chapter introduces the basic concepts and classification of PDEs, along with solution methods for the three classical equations.

Initial value problems give us complete information at the starting point, while boundary value problems provide partial information at both ends — like knowing the departure and destination points without knowing the complete path in between. These problems are extremely common in engineering: beam deflection, steady-state heat conduction, and eigenstate problems in the Schr ö dinger equation. This chapter explores in depth the theory and numerical methods for solving boundary value problems.

Many differential equations arising in practical problems cannot be solved by analytical methods — this is when numerical methods become our most powerful weapon. From Euler's simple idea to modern adaptive algorithms, numerical methods allow us to "approximately" solve virtually any differential equation. This chapter explores in depth the principles, implementations, and error analysis of various numerical methods.