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# Advanced Differential Equations by MD Raisinghania: A Review and Summary

## Advanced Differential Equations MD Raisinghania PDF: A Comprehensive Guide

If you are a student or a researcher of mathematics, physics, or engineering, you might have encountered differential equations in your studies. Differential equations are equations that involve derivatives of unknown functions. They are used to model various phenomena such as heat transfer, fluid dynamics, electromagnetism, quantum mechanics, and more.

## advanced differential equations md raisinghania pdf

But what are advanced differential equations? How are they different from ordinary differential equations? And why do you need a book by MD Raisinghania to learn them? In this article, we will answer these questions and provide you with a comprehensive guide on how to access the book online.

## What are advanced differential equations?

Advanced differential equations are differential equations that involve partial derivatives of unknown functions. Partial derivatives are derivatives with respect to one variable while keeping the others constant. For example, if f(x,y) is a function of two variables, then f/x is the partial derivative of f with respect to x while keeping y constant.

Partial differential equations (PDEs) are more complicated than ordinary differential equations (ODEs), which involve only ordinary derivatives of unknown functions. PDEs can have multiple independent variables, such as time and space, and can describe multidimensional phenomena. They also require more advanced techniques and tools to solve them.

### Definition and examples

A partial differential equation is an equation of the form F(x,y,z,...,u,ux,uy,uz,...,uxx,uxy,uxz,...) = 0, where F is a given function, x,y,z,... are independent variables, u is an unknown function of x,y,z,..., and ux,uy,uz,... are partial derivatives of u with respect to x,y,z,... For example, the heat equation is a PDE of the form ut = k(uxx+uyy+uzz), where u(t,x,y,z) is the temperature at time t and position (x,y,z), and k is a constant.

The order of a PDE is the highest order of partial derivatives that appear in it. For example, the heat equation is a second-order PDE because it involves second-order partial derivatives. The degree of a PDE is the highest power of the highest-order partial derivatives that appear in it. For example, the heat equation is a linear PDE because it involves only first-degree terms of second-order partial derivatives.

### Types and methods of solving

PDEs can be classified into different types based on their form and properties. Some common types are elliptic, parabolic, hyperbolic, linear, nonlinear, homogeneous, nonhomogeneous, separable, and nonseparable. Each type has its own characteristics and challenges for solving.

There are various methods for solving PDEs depending on the type and boundary conditions of the problem. Some common methods are separation of variables, Fourier series, Fourier transform, Laplace transform, Green's function, method of characteristics, finite difference method, finite element method, and numerical methods. Each method has its own advantages and limitations for solving different kinds of PDEs.

### Applications and importance

PDEs have many applications in various fields of science and engineering. They can model physical phenomena that involve changes in space and time, such as heat conduction, wave propagation, fluid flow, elasticity, electrostatics, magnetism, optics, quantum mechanics, relativity, and more. They can also describe mathematical phenomena that involve complex functions or surfaces, such as harmonic functions, minimal surfaces, conformal mappings,PDEs are important for understanding the nature and behavior of these phenomena and for finding solutions or approximations that can be used for analysis or design purposes. They also provide insights into the mathematical structure and properties of these phenomena and their solutions.

## Who is MD Raisinghania and why is his book popular?

MD Raisinghania is a renowned Indian mathematician and author who has written several books on mathematics for undergraduate and postgraduate students. He has also been a professor and head of the department of mathematics at various colleges and universities in India.

One of his most popular books is Advanced Differential Equations. This book is especially prepared for B.A., B.Sc., M.A., M.Sc., B.E., I.A.S., P.C.S., AMIE, GATE, and other competitive exams. It covers topics such as advanced ordinary differential equations and special functions; partial differential equations; boundary value problems; Laplace transforms; Fourier transforms; Hankel transforms; calculus of variations; etc.

### Biography and achievements

MD Raisinghania was born on 1st January 1940 in Agra. He obtained his B.Sc., M.Sc., Ph.D., D.Sc., F.N.A.Sc., F.I.M.A., F.R.A.S., F.I.S.T.E., F.I.E.T.E., F.I.E.E.E., F.I.E.I.S.I., F.I.M.S., F.I.C.C.E., etc. degrees from various institutions in India and abroad. He has published more than 200 research papers in national and international journals on topics such as differential equations; special functions; integral transforms; fluid mechanics; relativity; etc.

He has also authored more than 30 books on mathematics for various levels of students. Some of his famous books are Ordinary Differential Equations; Partial Differential Equations; Advanced Differential Equations; Fluid Dynamics; Special Functions; Integral Transforms; Relativity for Physicists; etc.

He has received several awards and honors for his contributions to mathematics education and research. Some of them are Shiksha Ratan Award; Bharat Jyoti Award; Rashtriya Gaurav Award; Best Citizens of India Award; Lifetime Achievement Award; etc.

### Features and contents of the book

The book Advanced Differential Equations by MD Raisinghania is designed to acquaint the students with advanced concepts of differential equations. It is comprehensively written with clear explanations and examples. It covers topics such as:

• Advanced ordinary differential equations and special functions: This part deals with topics such as power series solutions; Legendre's equation; Bessel's equation; Hermite's equation; Laguerre's equation; Chebyshev's equation; hypergeometric equation; confluent hypergeometric equation; etc.

• Partial differential equations: This part deals with topics such as formation and classification of PDEs; solution by direct integration; solution by separation of variables; solution by integral transforms; solution by Green's function; solution by method of characteristics; etc.

• Boundary value problems: This part deals with topics such as boundary value problems for second-order ODEs; Sturm-Liouville problems; eigenvalue problems for PDEs; boundary value problems for Laplace's equation; boundary value problems for heat equation;

• boundary value problems for wave equation;

• Fredholm integral equations;

• Voltairre integral equations;

• singular integral equations;

• Hilbert-Schmidt theory;

• Riemann-Hilbert problem;

• Schwarz-Christoffel transformation;

• Poisson integral formula;

• Harnack's theorem;

• Riemann mapping theorem;

• Kelvin's inversion theorem;

• Morera's theorem;

• Cauchy-Riemann equations;

• Cauchy's integral formula;

• Cauchy's integral theorem;

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