Partial Differential Equation

1. Define partial differential equations and classify first-order equations.
2. Construct solutions for partial differential equations using various methods.
3. Apply Lagrange's method to solve quasi-linear equations.
4. Explain the concept of shock and conservation law.
5. Apply Cauchy's method of characteristic equations.
6. Classify second-order partial differential equations.
7. Solve problems using the Cauchy-Kowalevski theorem.

13 weeks, about 68 hours in total. Yours will differ.

1. Week 1Module 1:

   • Unit 1: Definitions and Equations · 4 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..

2. Week 2Module 1:

   • Unit 2: Application of IVP Conservation Law, Development of Shock · 4 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..

3. Week 3Module 2:

   • Unit 1: General First Order Equation and Cauchy Method of Characteristic · 4 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..

4. Week 4Module 2:

   • Unit 2: Types of Solution · 4 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..

5. Week 5Module 3:

   • Unit 1: Second Order P.D.E. Classifications · 4 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..

6. Week 6Module 3:

   • Unit 2: Transformation of Independent Variables · 4 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..

7. Week 7Module 4:

   • Unit 1: Cauchy Problem, Characteristics Problem and Fundamental Existence Theorem · 4 hours

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

8. Week 8Module 1:

   • Unit 1: Definitions and Equations · 4 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 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..

9. Week 9Module 2:

   • Unit 1: General First Order Equation and Cauchy Method of Characteristic · 4 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 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..

10. Week 10Module 3:

    • Unit 1: Second Order P.D.E. Classifications · 4 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 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..

11. Week 11Module 4:

    • Unit 1: Cauchy Problem, Characteristics Problem and Fundamental Existence Theorem · 4 hours

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

12. Week 12Assignments

    • TMA Assignments · 6 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..

13. Week 13Revision

    • Final Revision · 6 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..