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MTH422Sciences3 Unitsintermediate

Partial Differential Equation

This course introduces Partial Differential Equations (PDEs) to 400-level undergraduate mathematics students. It covers essential definitions, classifications of first and second-order PDEs, and methods for constructing solutions. Topics include quasi-linear equations, Lagrange's method, conservation laws, Cauchy's method of characteristics, and the Cauchy-Kovalevsky theorem. The course aims to deepen understanding of PDEs and their applications through calculations and examples.

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156h

Study Time

13

Weeks

12h

Per Week

advanced

Math Level

Course Keywords

Partial Differential EquationsLagrange MethodCauchy ProblemConservation LawCharacteristics

Course Overview

Everything you need to know about this course

Course Difficulty

Intermediate Level

Builds on foundational knowledge

65%

intermediate

∑

Math Level

Advanced Math

📖

Learning Type

Theoretical Focus

Course Topics

Key areas covered in this course

1

Essential Definitions

2

First Order Equations

3

Quasi-Linear Equations

4

Method of Lagrange

5

Cauchy Method of Characteristic

6

Second Order P.D.E. Classifications

7

Transformation of Independent Variables

Total Topics7 topics

Requirements

Knowledge and skills recommended for success

Calculus

Ordinary Differential Equations

Linear Algebra

💡 Don't have all requirements? Don't worry! Many students successfully complete this course with basic preparation and dedication.

Assessment Methods

How your progress will be evaluated (3 methods)

assignments

Comprehensive evaluation of course material understanding

Written Assessment

tutor-marked assessments

Comprehensive evaluation of course material understanding

Written Assessment

final examination

Comprehensive evaluation of course material understanding

Written Assessment

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Study Schedule Beta

A structured 13-week journey through the course content

Week

1

Module 1:

4h

Unit 1: Definitions and Equations

4 study hours

•Review essential definitions of PDEs, including order, linearity, and homogeneity.
•Solve problems classifying different types of PDEs.
•Apply Lagrange's method to solve quasi-linear equations.

Week

2

Module 1:

4h

Unit 2: Application of IVP Conservation Law, Development of Shock

4 study hours

•Study the application of PDEs to conservation laws.
•Understand the development of shock waves and their implications.
•Work through examples demonstrating the formation of shocks.

Week

3

Module 2:

4h

Unit 1: General First Order Equation and Cauchy Method of Characteristic

4 study hours

•Study general first-order equations and the Cauchy method of characteristics.
•Practice sketching and explaining Monge cones.
•Solve problems using the Cauchy method of characteristics.

Week

4

Module 2:

4h

Unit 2: Types of Solution

4 study hours

•Categorize different types of solutions for PDEs: complete, general, and singular.
•Learn methods for deriving complete solutions.
•Understand the meaning and implications of general and singular solutions.

Week

5

Module 3:

4h

Unit 1: Second Order P.D.E. Classifications

4 study hours

•Classify second-order PDEs based on the properties of their eigenvalues.
•Understand the importance of eigenvalues in determining PDE behavior.
•Study Tricomi's equation and its characteristics.

Week

6

Module 3:

4h

Unit 2: Transformation of Independent Variables

4 study hours

•Learn how to transform independent variables to simplify PDEs.
•Apply theorems related to the regular case of variable transformation.
•Solve hyperbolic equations using transformations.

Week

7

Module 4:

4h

Unit 1: Cauchy Problem, Characteristics Problem and Fundamental Existence Theorem

4 study hours

•Study the Cauchy problem and characteristic problem.
•Understand the strip condition and its significance.
•Explore the fundamental existence theorem.

Week

8

Module 1:

8h

Unit 1: Definitions and Equations

4 study hours

•Review Module 1 Units 1 & 2: Definitions, Equations, Conservation Law and Development of Shock
•Solve additional exercises on essential definitions and first-order equations.

Unit 2: Application of IVP Conservation Law, Development of Shock

4 study hours

•Review Module 1 Units 1 & 2: Definitions, Equations, Conservation Law and Development of Shock
•Solve additional exercises on the application of IVP conservation law and the development of shock.

Week

9

Module 2:

8h

Unit 1: General First Order Equation and Cauchy Method of Characteristic

4 study hours

•Review Module 2 Units 1 & 2: General First Order Equation and Cauchy Method of Characteristic, Types of Solution
•Solve additional exercises on general first order equation and Cauchy method of characteristic.

Unit 2: Types of Solution

4 study hours

•Review Module 2 Units 1 & 2: General First Order Equation and Cauchy Method of Characteristic, Types of Solution
•Solve additional exercises on types of solution.

Week

10

Module 3:

8h

Unit 1: Second Order P.D.E. Classifications

4 study hours

•Review Module 3 Units 1 & 2: Second Order P.D.E. Classifications, Transformation of Independent Variables
•Solve additional exercises on second order P.D.E. classifications.

Unit 2: Transformation of Independent Variables

4 study hours

•Review Module 3 Units 1 & 2: Second Order P.D.E. Classifications, Transformation of Independent Variables
•Solve additional exercises on transformation of independent variables.

Week

11

Module 4:

4h

Unit 1: Cauchy Problem, Characteristics Problem and Fundamental Existence Theorem

4 study hours

•Review Module 4 Unit 1: Cauchy Problem, Characteristics Problem and Fundamental Existence Theorem
•Solve additional exercises on Cauchy problem and characteristics problem.

Week

12

Assignments

6h

TMA Assignments

6 study hours

•Work on Tutor Marked Assignments (TMAs).
•Focus on applying concepts from Modules 1 and 2 to solve assignment problems.
•Review feedback from previous TMAs to improve understanding.

Week

13

Revision

6h

Final Revision

6 study hours

•Complete and submit all Tutor Marked Assignments (TMAs).
•Final review of all course materials and key concepts.
•Prepare for final examinations by practicing with sample questions.

This study schedule is in beta and may not be accurate. Please use it as a guide and consult the course outline for the most accurate information.

Course PDF Material

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Study Tips & Exam Preparation

Expert tips to help you succeed in this course

1

Create concept maps linking Module 1 definitions to solution methods.

2

Practice solving quasi-linear equations using Lagrange's method from Unit 3 weekly.

3

Review examples of shock development from Unit 2 and identify key factors.

4

Focus on understanding the geometric interpretation of Cauchy's method in Module 2.

5

Master the classification of second-order PDEs from Module 3, Unit 1.

6

Review all TMAs and focus on areas where marks were lost.

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