Differential Equations 2 (751873001) 微分方程式 （二）
General Information
- Begins ~ ends: February 23, 2026 ~ June 12, 2026
- Instructor: Pu-Zhao Kow
- Email: pzkow [at] g.nccu.edu.tw
- Office hour: Friday (12:10 ~ 13:00)
- Teaching Language: Chinese and English
- Lecture Notes: Note: The lecture note may update during the course.
References
- J. W. E. Boyce and R. C. DiPrima, Elementary differential equations and boundary value problems, John Wiley and Sons, Inc., Hoboken, NJ, 12th edition, 2022. MR0179403, Zbl:1492.34001
- P.-F. Hsieh and Y. Sibuya, Basic theory of ordinary differential equations, Universitext, Springer-Verlag, New York, 1999. MR1697415, Zbl:0924.34001, doi:10.1007/978-1-4612-1506-6
Prerequisite
- Real and complex analysis
Homeworks
- Homework 1: Return by March 20, 2026 (Friday) 23:59
- Homework 2: Return by March 27, 2026 (Friday) 23:59
- Homework 3: Return by April 10, 2026 (Friday) 23:59
- Homework 4: Return by April 24, 2026 (Friday) 23:59
- Homework 5: Return by May 8, 2026 (Friday) 23:59
Schedule
- The lectures are on Friday (09:10-12:00) at 志希070116.
| Time | Room | Activities |
|---|---|---|
| 27.2.2026 09:10-12:00 | 志希070116 | Week 1 - no class: compensation holiday (peace memorial day) |
| 6.3.2026 09:10-12:00 | 志希070116 | Week 2: Preliminaries |
| 13.3.2026 09:10-12:00 | 志希070116 | Week 3: Weak derivatives and distribution derivatives |
| 20.3.2026 09:10-12:00 | 志希070116 | Week 4: Definition and elementary properties of the Sobolev spaces [Return Homework 1 by 23:59] |
| 27.3.2026 09:10-12:00 | 志希070116 | Week 5: [Return Homework 2 by 23:59] |
| 3.4.2026 09:10-12:00 | 志希070116 | Week 6 - no class: compensation holiday (children’s day) |
| 10.4.2026 09:10-12:00 | 志希070116 | Week 7: Solving elliptic PDE for small wave number, the maximum principle [Return Homework 3 by 23:59]
click me to see the title and abstract of today's first talk
Speaker. Li, Bo-Jyun Title. An introduction of Ito's calculus Abstract. This talk explores the construction of the Ito integral, beginning with the transition from Riemann-Stieltjes integration to the stochastic domain. While classical integration requires the integrator to be of bounded variation, Brownian motion violates this condition, necessitating a new framework for stochastic integration. We construct the Ito integral starting from elementary functions and extend it to adapted, square-integrable processes through the fundamental Ito Isometry and the completeness of \(L^2\) spaces. This process establishes the integral as a martingale transform that maintains linearity and continuity. The discussion concludes by highlighting how the reliance on left-endpoint evaluation distinguishes Ito's calculus from deterministic frameworks, particularly through the emergence of a quadratic variation term in the stochastic integration-by-parts formula. |
| 17.4.2026 09:10-12:00 | 志希070116 | Week 8: Solving elliptic PDE: Eigenvalue problem and Fredholm alternative |
| 24.4.2026 09:10-12:00 | 志希070116 | Week 9: Existence of weak solution via Galerkin approximation [Return Homework 4 by 23:59] |
| 1.5.2026 09:10-12:00 | 志希070116 | Week 10 - no class: labor day |
| 8.5.2026 09:10-12:00 | 志希070116 | Week 11: Uniqueness of weak solution via Gronwall inequality, Fourier series [Return Homework 5 by 23:59]
click me to see the title and abstract of today's first talk
Speaker. Kao, An-Hsien Title. Restart and Deflation in Golub-Kahan Bidiagonalization Abstract. The Golub-Kahan Bidiagonalization (GKB) process faces two practical difficulties: progressive loss of orthogonality among Lanczos vectors, and prohibitive memory growth as the subspace expands. Reorthogonalization and thick restart address these issues by restoring orthogonality and limiting subspace size. When additional singular triplets are needed beyond an existing partial SVD, explicit deflation projects out already-converged singular vectors from the matrix-vector product, directing subsequent GKB iterations toward the remaining spectrum. |
| 15.5.2026 09:10-12:00 | 志希070116 | Week 12
click me to see the title and abstract of today's first talk
Speaker. Li, Bo-Jyun Title. An Introduction to SDEs and Financial Applications Abstract. This talk introduces the fundamentals of stochastic calculus and its applications in finance. Starting from the Fundamental Theorem of Calculus, we introduce the Itô formula and highlight its departure from classical calculus. We then transition from ODEs to Stochastic Differential Equations (SDEs), discussing the existence and uniqueness theorem via Picard's method under Lipschitz conditions, while clarifying the distinctions between strong and weak solutions. The session concludes with financial applications, specifically using the Itô formula to solve the Geometric SDE and establishing the foundation for the Black-Scholes model. click me to see the title and abstract of today's second talk
Speaker. Wu, Pei-Wei Title. Linear programming Abstract. This presentation introduces linear programming and its dual problem, and demonstrates how to solve them using the simplex method. |
| 22.5.2026 09:10-12:00 | 志希070116 | Week 13 - no class: Attending conference (2026 NCTS Workshop on Mathematics of Living Systems) |
| 29.5.2026 09:10-13:00 | 志希070116 | Week 14
click me to see the title and abstract of today's first talk
Speaker. Song, Jia-Ying Title. Convergence of the Backward Euler Method: ODE Version Abstract. This presentation explains why the Backward Euler method converges when used to approximate the exact solution of an ordinary differential equation (ODE). The proof also illustrates why the Backward Euler method is classified as a one-step method. click me to see the title and abstract of today's second talk
Speaker. Song, Jia-Ying Title. Convergence of the Backward Euler Method: PDE Version Abstract. This presentation explains why the Backward Euler method converges when used to approximate the exact solution of a partial differential equation (PDE). It also discusses why weak convergence is needed in the proof, how it is applied, and concludes with a simple example. click me to see the title and abstract of today's third talk
Speaker. Chan, Yung-Hsiang Title. Wave equation I Abstract. This presentation will introduce 1-dimensional wave equation on the whole line \(\mathbb{R}\) and solve the general solution. |
| 5.6.2026 09:10-13:00 | 志希070116 | Week 15
click me to see the title and abstract of today's first talk
Speaker. Kao, An-Hsien Title. Importance Sampling for SRAM Failure Estimation: A SVM-Based Approach to Shift Selection Abstract. Estimating rare failure probabilities in SRAM cells requires methods that go beyond standard Monte Carlo simulation. In this talk, we introduce two common importance sampling strategies — mean shifting and variance scaling — and motivate the use of mean shifting for SRAM failure analysis. A key challenge in this approach is the selection of an appropriate shift vector. We present an SVM-based method to learn the pass/fail boundary and identify effective shift vectors more efficiently, reducing the number of required simulations in the calibration phase while maintaining estimation accuracy. click me to see the title and abstract of today's second talk
Speaker. Lin, Pei-Syuan Title. Denoising Diffusion Probabilistic Models Abstract. In this talk, we introduce Denoising Diffusion Probabilistic Models (DDPM), walking through the forward noising process, the reverse denoising step. click me to see the title and abstract of today's third talk
Speaker. Chan, Yung-Hsiang Title. Wave equation II Abstract. This presentation will introduce 1-dimensional wave equation on the half-line \((0,\infty)\) and finite interval \((0,L)\) and show how to solve the equation. click me to see the title and abstract of today's fourth talk
Speaker. Wu, Pei-Wei Title. Rook polynomial Abstract. This presentation will introduce restricted position problem and rook polynomial. |
| 12.6.2026 09:10-13:00 | 志希070116 | Week 16
click me to see the title and abstract of today's first talk
Speaker. Lin, Pei-Syuan Title. Diffusion Posterior Sampling for General Noisy Inverse Problems Abstract. In this talk, we will extend the discussion of Denoising Diffusion Probabilistic Models (DDPM), using Diffusion Posterior Sampling (DPS) as a case study to show how a pre-trained diffusion model can be applied to certain image inverse problems (such as deblurring, super-resolution, and inpainting) without retraining, along with the core intuition behind the guidance term and how it works. click me to see the title and abstract of today's second talk
Speaker. Hung, Chun-Wei Title. Shannon entropy Abstract. This presentation will introduce Shannon entropy which quantify uncertainty of a random variable. Mutual information and KL divergence will also be introduced to understand the relationship between two models. click me to see the title and abstract of today's third talk
Speaker. Hung, Chun-Wei Title. Transfer entropy Abstract. This presentation will introduce transfer entropy, a method used to analyze time series that can answer a more dynamic question: if we know the past of one system, can it help us better predict the future of another system? |
Completion
- The course can be taken for credit by attending the lectures, returning written solutions (60%) in \(\LaTeX\) and giving (at least) 2 presentations (each 20%).