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所在平台: Udemy |
课程主页: https://www.udemy.com/course/partial-differential-equations-tutorial-college-mathematics/
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**课程名称：** 理解用于人工智能的偏微分方程 **课程概述：** 本课程旨在为学习者提供偏微分方程（PDE）的全面介绍，重点关注其在人工智能领域的应用。课程内容涵盖了 PDE 的基础知识，包括： * **PDE 导论：** 解释什么是齐次 PDE（一度和二度），以及如何形成 PDE（讨论了 $f(u,v)=0$ 形式的方程）。 * **链式法则：** 讲解当 $z$ 表示为两个函数之和时的链式法则应用。 * **一阶 PDE：** 深入探讨一阶偏微分方程的求解方法。 * **求解 PDE：** * 教授如何通过直接积分求解非齐次 PDE。 * 介绍求解齐次 PDE 的步骤，目标是成为一个理想的 PDE 求解器。 * **数学趋势：** 强调 PDE 研究在学术研究中的重要性和长远价值。 * **课程要点：** * 区分齐次和非齐次 PDE 概念。 * 学习如何求解不同类型的 PDE。 * 掌握形成 PDE 的技巧。 * 提供所有课程公式的汇总。 * 包含 PDE 示例讲解。 * **进阶内容：** * **极坐标：** 介绍极坐标系及其与直角坐标系的关系，以及相关基础公式。 * **高阶导数：** 讲解如何处理二阶以上（包括 $n$ 阶）导数的计算。 * **非线性微分方程：** 介绍非线性微分方程及其求解方法。 * **精确微分方程：** 提供精确微分方程的基础理解。 * **曲线的垂直轨迹：** 探索曲线的垂直轨迹概念。 * **牛顿冷却定律：** 运用 PDE 解释牛顿冷却定律。 * **逆微分算子：** 学习逆微分算子的概念及其在问题解决中的应用。 * **约化公式：** 掌握约化公式的原理及其在求解问题中的使用。 **先修知识：** 具有一定的线性微分方程（常系数）求解知识会有帮助，课程中也会在“重要技巧和公式”的讲座中进行复习。 **课程收获：** 通过本课程，学习者将获得解决 PDE 的扎实基础知识，为进一步在人工智能等领域应用 PDE 打下坚实基础。课程最后设有作业，以巩固所学内容。
Solving PDE's can be classified into the following partsIntroduction to Partial Differential EquationsYou learn what is a homogeneous PDE, Both degree 1 and 2 are discussed hereFormation of PDE is taught , equation of the form f(u,v) =0 is discussed. Chain Rule is also taught when z is expresses as the sum of 2 functions. First order partial differential equations are discussed.PDE math teaches you how to solve a non homogeneous partial differential equation by direct integration.Steps to solve a homogeneous PDE is also discussed. An ideal partial differential equations solver.What trends are occurring in Mathematics? The study of Partial Differential equations is definitely one which will take you long in your field of study. This course introduces you to PDE, explains the difference between homogeneous and non homogeneous equations and how to solve each of them. You are also taught techniques on how to form a PDE. A simple course which promises you a depth of knowledge! A little knowledge of solving linear differential equations with constant coefficients is needed. This is discussed in my lecture on important tips and formulas. There is an assignment at the end. All the formulas learnt in this course are also compiled together for you.Don't miss out on partial differential equations examples.Learn about polar coordinates and the relationship between rectangular and polar coordinates. Also learn basic formulas for these.Also learn how to go beyond second order derivatives and evaluate nth order derivatives. You'll be introduced to non linear differential equations and how to solve them. Get a basic understanding of Exact Differential equations.Explore Orthogonal Trajectories of a Curve. Newton's Law of Cooling is also explained here. Learn about the Inverse Differential Operator and how to use it in problem solving.Understand the basics of the Reduction formula and how to use it in problem solving.
