# PARTIAL DIFFERENTIAL EQUATIONS Nisha V M
Mathematics
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# ME010302-Partial Differential Equations

Course Code: ME010302                                         Coordinator: Nisha V M

Course Credit: 4

Semester: III

Course Type: Core

Department: Mathematics

### COURSE OBJECTIVES:

1. Introduce the difference between Ordinary differential equation and partial differential equations
2. Introduce students to the origin and formation of partial differential equations.
3. Introduce students to how to solve linear and non linear Partial Differential equations using different methods.

### LEARNING OUTCOMES:

On successful completion of this course students will be able to:

1. solve linear partial differential equations of both first and second order
2. apply specific methodologies, techniques and resources to conduct research and produce innovative results in the area of specialisation.
3. identify real phenomena as models of partial derivative equations.
4. extract information from partial derivative models in order to interpret reality.
5. Apply partial derivative equation techniques to predict the behaviour of certain phenomena.
6. Analyse, synthesise, organise and plan projects in the field of study

### SYLLABUS:

Module 1: (20 hours)

Methods of solutions of dx/P = dy/Q =  dz/R.  Orthogonal trajectories  of a  system of curves on a surface. Pfaffian differential forms and equations. Solution of Pfaffian differential equations in three variables, Partial differential equations. Origins of first order partial differential equation.

Module 2:      (25 hours)

Linear equations of first order.  Integral  surfaces  passing  through  a  given curve. Surfaces orthogonal to a given system of surfaces. Nonlinear partial differential equation of the first order . Compatible systems of first order equations . Charpits Method. Special types of first order equations. Solutions satisfying given conditions.

Module 3:    (20 hours)

Jacobi’ s method The origin of second order  equations.  Linear  partial differential equations with constant coefficients. Equations with variable coefficients.

Module 4.:  (25 hours)

Separation of variables. Non linear equations of the second order . Elementary solutions of Laplace equation. Families of equipotential surfaces. The two dimensional Laplace Equation Relation of the Logarithmic potential to the Theory of Functions.

## Text Book:

Ian Sneddon , Elements of Partial Differential Equations, Mc Graw Hill Book Company

## References:

1. Phoolan Prasad and Renuka Ravindran, Partial Differential Equations, New Age International
2. K Sankara Rao, Introduction to Partial Differential Equations, Prentice Hall of India
3. E T Copson, Partial Differential Equations, S Chand and Co

### TEACHING SCHEDULE:

 Topics Learnig Outcomes Hours Module I 1 Methods of solutions of dx/P = dy/Q =  dz/R Learn to solve a differential equation of the form dx/P = dy/Q =  dz/R 4 2 Orthogonal trajectories  of a  system of curves on a surface Able to find the orthogonal trajectories of a system of surfaces 3 3 Pfaffian differential forms and equations Identify Pfaffian differential equation  and its properties 4 4 Solution of Pfaffian differential equations in three variables Different methods to solve Pfaffian differential equations 5 5 Partial differential equations Distinguish between ODE and PDE 2 6 Origins of first order partial differential equation How to form a PDE from a given function by eliminating arbitrary constats and functions. 2 Module II 1 Linear equations of first order Identify linear partial differential equations and to solve such an equation 3 2 Integral  surfaces  passing  through  a  given curve To find the integral surface passes through a given curve 3 3 Surfaces orthogonal to a given system of surfaces To find an equation of the surface which is orthogonal to a given system of surfaces 3 4 Nonlinear partial differential equation of the first order Identify a non linear partial differential equation 3 5 Compatible systems of first order equations Understand the term compatible in case of PDE 4 6 Charpits Method Solve PDE using charpits method 3 7 Special types of first order equations Under some special types of PDE and its solutions 3 8 Solutions satisfying given conditions. Able to find a solution of a PDe which satisfy some conditions 3 Module III 1 Jacobi’ s method Learn to solve PDE using Jacobi’s method 4 2 The origin of second order  equations Identify a second order PDE 2 3 Linear  partial differential equations with constant coefficients Learn to solve linear PDE with constant coefficients 8 4 Equations with variable coefficients. Learn to solve PDE with variable coefficients 6 Module IV 1 Separation of variables Learn to fins the solution of second order on linear PDE by separation of variables 5 2 Non linear equations of the second order Learn more methods to solve second order non linear PDE 6 3 Elementary solutions of Laplace equation Learn about Laplace equation and its solution 4 4 Families of equipotential surfaces. Under the term equipotetial surfaces and solve problems 4 5 The two dimensional Laplace Equation Learn about two dimensional wave equations and its solutions 4 6 Relation of the Logarithmic potential to the Theory of Functions Identify the relation of the Logarithmic potential to the Theory of Functions 3 90

EVALUATION STRATEGY:

The internal evaluation is based on the participant’s attendance, class participation, assignments, seminars and internal assessment and continuous evaluation tests

• ##### Module I

Methods of solutions of dx/P = dy/Q =  dz/R.  Orthogonal trajectories  of a  system of curves on a surface. Pfaffian differential forms and equations. Solution of Pfaffian differential equations in three variables, Partial differential equations. Origins of first order partial differential equation.

No items in this section
• ##### Module II

Linear equations of first order.  Integral  surfaces  passing  through  a  given curve. Surfaces orthogonal to a given system of surfaces. Nonlinear partial differential equation of the first order . Compatible systems of first order equations . Charpits Method. Special types of first order equations. Solutions satisfying given conditions.

• ##### Module III

Jacobi’ s method The origin of second order  equations.  Linear  partial differential equations with constant coefficients. Equations with variable coefficients.

No items in this section
• ##### Module IV

Separation of variables. Non linear equations of the second order . Elementary solutions of Laplace equation. Families of equipotential surfaces. The two dimensional Laplace Equation Relation of the Logarithmic potential to the Theory of Functions.

No items in this section Nisha V M
Nisha V M, Assistant Professor ,Department of Mathematics, St Paul's College ,Kalamassery. I have joined here in 2015. I have done my PG and MPhil in Mathematics from CUSAT, Kalamassery. My area of interest is Number theory and Graph Theory.

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