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所在平台: Udemy |
课程主页: https://www.udemy.com/course/numerical-methods-and-series-solution-of-equations/
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**计算数值分析在人工智能和数据科学中的应用** 本课程面向计算机科学、数学和工程学专业的学生，旨在深入介绍人工智能和数据科学领域中至关重要的数值分析方法。 **课程内容概览：** 1. **泰勒级数法及其改进：** * 学习泰勒级数法的基本原理，并掌握其如何用于求解方程到小数点后四位。 * 进一步学习改进的欧拉定理，了解其作为泰勒级数法改进之处，以及如何应用于方程求解。 2. **高阶龙格-库塔法及预测-校正方法：** * 深入学习四阶龙格-库塔法，并通过实例演示其在解决问题中的应用。 * 掌握两类重要的预测-校正方法：米尔恩（Milne's Predictor）方法和亚当斯-巴弗斯（Adam Bashforth）方法，并理解其迭代计算过程。 3. **贝塞尔微分方程及其解法：** * 学习贝塞尔微分方程的推导过程，并详细讲解其求解步骤，最终引出贝塞尔函数。 * 了解伽马函数在贝塞尔函数中的应用。 4. **贝塞尔函数的性质：** * 探索贝塞尔函数的多种重要性质，并详细证明其正交性，从而引出两种特殊情况，其中一种将导向洛姆梅尔积分公式。 5. **勒让德微分方程与勒让德函数：** * 学习使用幂级数方法求解勒让德微分方程。 * 掌握勒让德函数的推导过程，以及勒让德多项式如何导向罗德里格公式。 6. **数值分析中的其他重要方法：** * **有限差分法：** 学习有限差分的概念、前向差分和后向差分表，以及如何在问题求解中应用它们。 * **假位法（Regula Falsi Method）：** 学习使用假位法解决数值分析问题。 * **牛顿-拉夫逊法（Newton Raphson's Method）：** 掌握如何使用牛顿-拉夫逊法加速方程求解。 * **数值积分：** 学习应用辛普森（Simpson's）一/三规则、辛普森三/八规则以及韦德尔（Weddle's）规则进行数值积分。 * **高阶导数的计算：** 学习计算函数n阶导数的方法及其评估。 7. **问题解决与应用：** * 通过实际问题，巩固对罗德里格公式的应用。 * 课程最后将通过一个综合性作业，回顾课程中的关键概念和解题技巧。 **课程特色：** * **理论与实践结合：** 涵盖详细的理论推导和实际问题应用。 * **循序渐进：** 从基础概念到复杂方法，层层递进。 * **数学提升：** 旨在帮助学习者更好地理解和掌握数学知识，并建立学习数学的信心。 **学习目标：** 通过本课程的学习，学员将能够熟练运用各种数值分析方法解决复杂问题，为他们在人工智能和数据科学领域的进一步学习和研究打下坚实的基础。
Welcome to this course on Numerical Methods and Series solutions of Differential Equations. This course is primarily intended for you if you are studying Math in College and if you are learning Engineering Math. Tips to help you understand Math better.You will start with a brief introduction to Taylor Series method and how to use Taylor Series method to solve an equation upto 4 decimal places. As a modification of Taylor series method, you will learn modified Euler's theorem and how to use it to solve equations. As a side note, these formulas involve a lot of calculations at the problem solving stage.Next, you will be introduced to Runge Kutta method of 4th order and how to use it in solving problems. 2 predictor and corrector methods are taught here, namely Milne' s Predictor method and Adam Bashforth methods. The numericals here involve several iterations and have been explained step by step.In lesson 3, you will be introduced to Bessel's Differential equation and how to solve it. The solution is rather lengthy and has been explained keeping all steps in mind. This will lead you to the Bessel's function at the end. Note the use of Gamma functions in Bessel's function is shown.Lesson 4 is on Bessel's function and it's properties. The orthogonality property of the Bessel's function is also proved leading to two cases, one of which leads to Lommel's Integral formula. In Lesson 5, you will learn about Legendre Differential equation and how to solve it using the power series method. As a conclusion to this , you will learn about Legendre functions and how they are derived from Legendre Differential Equations. How Legendre Polynomials lead to Rodrigue's formula is also shown.Lesson 6 is a problem solving session where you will learn to use Rodrigue's formula to solve problems.The course concludes with an assignment which discusses possible questions that can be asked. Note that each of these questions have been discussed during the course.You'll also learn what is nth order derivative of a function and how to evaluate it. An interesting and important topic.Learn the method of finite differences and the forward and backward difference table and how to use it in problem solving. Learn how to solve problems using the Regula Falsi Method in Numerical Analysis.Learn how to solve equations faster using the Newton Raphson's Method. You'll learn how to apply Simpson's one third Rule, Simpson's three by eighth rule and Weddle's Rule in Numerical Integration. An important point to keep in mind is that this course is highly theoretical and involves you to write these proofs to gain mastery.Motivating you to learn Mathematics!
