
AU Mathematics III Sem III Unit 3 : Applications of Partial Differential Equations(Method of separation of variables, Vibrations of a stretched stringwave equations, Onedimensional and twodimensional heat flow equations, Solution of Laplace s equation, Laplace s equation in polar coordinates) 


1. Situations giving rise to Partial Differential Equations(Mathematics) 
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A mass balance is an application of conservation of mass to the analysis of physical systems. By accounting for material entering and leaving a system, mass flows can be identified which might have been unknown or difficult to measure without this technique.
Title: Math 3.4 Mathematical Modeling




2. Mathematical Modelling for Convection Diffusion and Reaction Processes(Mathematics) 
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After studying this unit, students will be able to analyse about convection diffusion processesBurger's equation which will cover Burger's equation and the plane wave solution, ColeHopf transformation and the exact solution of Burger's equation. They will also be able to explain asymptotic behavior of the exact solution of Burger's equation and Burger's initial and boundary value problem and exp
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Title: Math 3.4 Mathematical Modeling




3. Setting up of First Order Differential Equations and their Solutions(Mathematics) 
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Differential equations occur quite frequently in our daily life. The motion of an object can always be associated with a differential equation. The change in prices of commodities, the flow of fluids, the concentration of chemicals etc., often lead to differential equations. Such equations may depend on one or more independent variables. Further, it may include the derivatives of the first or high
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Title: Math 3.4 Mathematical Modeling




4. Variational Principles Probability Generating Function and Model For Traffic Flow(Mathematics) 
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A variational principle is a scientific principle used within the calculus of
variations, which develops general methods for finding functions which minimize or maximize the value of quantities that depend upon those functions.
Title: Math 3.4 Mathematical Modeling




5. Introduction to PDE(Mathematics) 
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In this unit, the reader studies about introduction and formation of partial differential equation (PDE). They also study the classification of first order PDE and some basic definition pertaining to POE.
Title: Math 1.5 Differential Equations




6. Mathematical Modeling of Some Fundamental Problems(Mathematics) 
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Mathematical Modeling in terms of differential equations arises when the situation modeled involves some continuous variables varying with respect to other continuous variables and we have reasonable hypothesis about the rate of change of dependent variables with respect to independent variables. Mathematical models in terms of ordinary differential equations will be studied in this unit and the u
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Title: Math 3.4 Mathematical Modeling




7. Cauchy Problem(Mathematics) 
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In this unit, the reader studies about the Cauchy problem, principle of superposition. They also study the method of characteristics to solve semilinear, quasilinear and nonlinear PDEs.
Title: Math 1.5 Differential Equations




8. Introduction to mathematical Modeling(Mathematics) 
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A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences such as physics, biology, earth science, meteorology and engineering disciplines like computer science, artificial intelligence, but also in the social sciences Physi
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Title: Math 3.4 Mathematical Modeling




9. Complete Integrals of PDE(Mathematics) 
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In this unit, the reader studies various methods such as Charpit's method, Jacobi's method for finding the solution (Complete integral) of POE.
Title: Math 1.5 Differential Equations




10. Limitations of Mathematical Modeling(Mathematics) 
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A crucial part of the modeling process is the evaluation of whether or not a given mathematical model describes a system accurately. This question can be difficult to answer as it involves several different types of evaluation. In this regard we have discussed the limitations of modeling in following sections.
Title: Math 3.4 Mathematical Modeling




11. Existence of Solutions of ODE(Mathematics) 
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In this unit, the reader will learn some basic definitions of ordinary differential equations. Also the reader will study the existence and uniqueness of solutions of initial value problem and a system of differential equations. In order to fully understand the proofs of these theorems, the reader must be familiar with certain concepts of real function theory.
Title: Math 1.5 Differential Equations




12. Ordinary Differential Equations and Finite Difference Methods(Mathematics) 
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This unit explains Initial Value Problem and linear second order differential equations. It also explains nonlinear second order differential equations. and describes finite difference methods.
Title: Math 2.4 Numerical Analysis




13. PredictorCorrector Methods and Stiff System(Mathematics) 
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We shall discuss the application of the explicit and implicit multistep methods for the solution of the initial value problems in this unit.
Title: Math 2.4 Numerical Analysis




14. Classification of Second Order PDE(Mathematics) 
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In this unit the reader studies classification of second order partial differential equation. They also study the method of characteristics and classify the given second order POE to its respective canonical form.
Title: Math 1.5 Differential Equations




15. Eigen function and Green's Function(Mathematics) 
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In this unit the reader studies about the Eigenfunctions. Eigenfunction expansion formula. Green's function.
Title: Math 1.5 Differential Equations




16. Solutions to NonLinear System of ODE(Mathematics) 
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The objective of this unit is the study of Nonlinear Autonomous system of ordinary differential equations, critical points, path and their nature. The reader also studies the method to find the nature of critical points. stability and paths by Liapunov direct method.
Title: Math 1.5 Differential Equations




17. Special Functions(Mathematics) 
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In this unit, the reader studies the power series solution by Frobenius method for Laguerre equation, Hermite differential equation, Chebyshev equation, hypergeometric equation.
Title: Math 1.5 Differential Equations




18. Solutions to Linear system of ODE(Mathematics) 
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The main objective of this unit is the study of linear system of ordinary differential equations.
Title: Math 1.5 Differential Equations








19. Methods of Solving ODE 2(Mathematics) 
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In this unit the reader studies the zeros of solutions of ODE. Sturm's separation theorem. Sturm's comparison theorem , SturmLiouville problems. The reader also study the method of variation o f parameters. Eigenvalue problems.
Title: Math 1.5 Differential Equations




20. Methods of Solving ODE  1(Mathematics) 
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In this unit, the reader first learn the definitions of linearly independent and linearly dependent set of functions. Then they study the fundamental set of solutions for ordinary differential equations, Wronskian, Adjoint differential equations and Green's formula.
Title: Math 1.5 Differential Equations




21. Power Series Solution of ODE(Mathematics) 
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In this unit, the reader studies the power series solution o f linear equations of second order around ordinary and regular singular points.
Title: Math 1.5 Differential Equations




22. The Laurent Series, Partial Fractions,MittagLeffler's Theorem(Mathematics) 
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In this unit, we discuss the infinite product and their relations with infinite series. Further, we study Weierstrass Factorization theorem which deals with representation of meromorphic function as an infinite product. We also study canonical product.
Title: Math 2.3 Complex Analysis  II




23. Analytic Functions, CauchyRiemann Equations, Harmonic Functions, Rational Functions(Mathematics) 
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The objective of this unit to show that a function of a complex variable can have a derivative only if the real and imaginary parts satisfy a system of a partial differential equations called the CauchyRiemann equations. Further study of the CauchyRiemann equations shows a connect ion between complex analysis and harmonic functions.
Title: Math 1.3 Complex Analysis  I




24. Difference Equations and Generating Functions(Mathematics) 
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After going through this unit we can define and construct generating functions for sequences arising in various types of combination problems. Also we use generating functions to find the number of integer solution to linear equations. We solve recurrence relation using generating functions.
Title: Math 1.4 Discrete Mathematics




25. Bisection Method and Iterative Methods of Transcendental Equations (Mathematics) 
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This unit deals with the definition of transcendental function and explains the Bisection Method, Secant and RegulaFalsi Methods and Newton Raphson Method. It also explains Muller method, Chebyshev method and describes the multipoint iteration methods to find roots of the equations.
Title: Math 2.4 Numerical Analysis






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