微分方程

課程將提供給學生完整的微分方程觀念介紹。加強學生對原微分方程的了解並引起興趣。課程的目標在幫助學生了解微分方程式,其解的意義,以及微分方程式在研究和模擬上的應用。 Differential equations for researchers are very important tools. This course is designed to provide students with a comprehensive introduction to concepts and ideas that form the basis of solutions of the differential equations. The course will enhance students’ understanding of and interest on the ordinary differential equations. The course is aiming to help students to understand differential equations, what the solutions of the equations mean, and how differential equations can be used to answer pertinent questions in modeling and research.

微積分/線性代數 Calculus/Linear Algebra

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Homework:10% Midterm Exam:30% *2 Final Exam:30%

• First-order differential equations

  Integrals as general and particular solutions. Slope fields and solution curves. Existence and uniqueness of solutions. Separable equations. Linear first-order equations. Substitution methods and exact equations. Population models. Cooling and heating. Acceleration-velocity models.

• Linear equations of higher order

  Homogeneous linear equations. Principle of superposition. Existence and uniqueness of solutions. Linear dependence (independence) for a set of functions. Wronskian. General solutions. Solution of homogeneous linear equations with constant coefficients. Characteristic equation and its roots. Non-homogenous linear equations and the method of undetermined coefficients. Variation of parameters. Forced oscillation and resonance. Electrical circuits. Endpoint problems, eigen-values and eigen-functions.

• Laplace transform methods

  Laplace transforms and inverse transforms. Existence of Laplace transforms. Transforms of derivatives (including piecewise continues functions) and integrals. Transformation of initial value problems. Translation and partial fractions. Derivatives, integrals, and products of transforms (convolution of two functions). Periodic and piecewise continuous input functions (unit step function). Transforms of impulse and delta functions (as operators). Duhamel’s principle.

• Linear systems of differential equations

  Systems of differential equations and applications. The method of elimination. Review of matrices and linear systems. First order linear systems of differential equations. Principle of superposition, independence, Wronskians, and general solutions. The eigen-value method for homogeneous system. Second-order systems and mechanical applications. Multiple eigen-value solutions. Matrix exponentials. Non-homogeneous linear systems.

• Power series methods

  Review of power series. Series solutions of ordinary homogenous linear second order equations near ordinary points and regular singular points. Method of Frobenius (including the exceptional cases). Bessel equation and its solutions.

• Fourier series methods

  Periodic function and trigonometric Fourier series. General Fourier series and convergence. Fourier sine and cosine series. Term-wise differentiation and integration of Fourier series. Applications of Fourier series. Examples of partial differential equations (heat conduction, wave equation). Boundary conditions, separation of variables. Laplace’s equation. Dirichlet problems.

• Eigen-values and boundary value problems (Introduction)

  Sturm-Liouville problems. Orthogonality of eigen-functions and eigen-function expansions. Applications of eigen-function series.

C. H. Edwards and D. E. Penney Elementary Differential Equations with Boundary Value Problems 6th edition, 2014, Pearson Education Limited

地點

工程四館408室

時間

週二10:00~12:00

聯絡方式

TEL:03-5131289 Email:graylin@mail.nctu.edu.tw