 50 students
 1 lessons
 0 quizzes
 10 week duration
ME010302Partial Differential Equations
Course Code: ME010302 Coordinator: Nisha V M
Course Credit: 4
Semester: III
Course Type: Core
Department: Mathematics
COURSE OBJECTIVES:
 Introduce the difference between Ordinary differential equation and partial differential equations
 Introduce students to the origin and formation of partial differential equations.
 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:
 solve linear partial differential equations of both first and second order
 apply specific methodologies, techniques and resources to conduct research and produce innovative results in the area of specialisation.
 identify real phenomena as models of partial derivative equations.
 extract information from partial derivative models in order to interpret reality.
 Apply partial derivative equation techniques to predict the behaviour of certain phenomena.
 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:
 Phoolan Prasad and Renuka Ravindran, Partial Differential Equations, New Age International
 K Sankara Rao, Introduction to Partial Differential Equations, Prentice Hall of India
 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.

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.

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.

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