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Linear Integral Equations By Shanti Swarup Pdf Download




If you are looking for a comprehensive and accessible book on linear integral equations, you might want to check out Linear Integral Equations by Shanti Swarup. This book is a classic text that covers both theory and applications of this important branch of mathematics. In this article, we will give you an overview of what linear integral equations are, why they are important, who is Shanti Swarup and what is his book about, and how to download the pdf version of the book.




Linear Integral Equations By Shanti Swarup Pdf Download



What are linear integral equations?




Linear integral equations are a type of mathematical equations that involve integrals of unknown functions. They have the general form:


$$\int_a^b K(x,t) f(t) dt = g(x)$$


where K(x,t) is a given function called the kernel, f(t) is the unknown function to be determined, g(x) is a given function called the source term, and a and b are constants.


Definition and examples




An integral equation is said to be linear if both the unknown function f(t) and its integral appear linearly in the equation. That means that if f(t) and g(t) are solutions of the equation, then so is any linear combination of them, such as c1f(t) + c2g(t), where c1 and c2 are constants.


Some examples of linear integral equations are:


  • Fredholm equation of the first kind: $$\int_a^b K(x,t) f(t) dt = g(x)$$ where K(x,t) is continuous.



  • Fredholm equation of the second kind: $$f(x) + \lambda \int_a^b K(x,t) f(t) dt = g(x)$$ where K(x,t) is continuous and $\lambda$ is a constant.



  • Volterra equation of the first kind: $$\int_a^x K(x,t) f(t) dt = g(x)$$ where K(x,t) is continuous.



  • Volterra equation of the second kind: $$f(x) + \lambda \int_a^x K(x,t) f(t) dt = g(x)$$ where K(x,t) is continuous and $\lambda$ is a constant.



  • Singular integral equation: $$\int_a^b \fracK(x,t)x-t f(t) dt = g(x)$$ where K(x,t) has a singularity at x=t.



Types and methods of solving




Linear integral equations can be classified into two types: homogeneous and non-homogeneous. A homogeneous equation has g(x)=0 for all x, while a non-homogeneous equation has g(x)$\neq$0 for some x. The solution of a homogeneous equation is either trivial (f(t)=0 for all t) or non-trivial (f(t)$\neq$0 for some t). The solution of a non-homogeneous equation can be obtained by adding a particular solution (a solution that satisfies the equation for any g(x)) and a general solution (a solution that satisfies the homogeneous equation).


There are various methods of solving linear integral equations, depending on the type and form of the equation. Some of the common methods are:


  • Direct methods: These methods involve transforming the integral equation into an equivalent system of linear algebraic equations, which can be solved by matrix methods. Examples of direct methods are quadrature methods, collocation methods, Galerkin methods, etc.



  • Iterative methods: These methods involve finding successive approximations to the solution by using an initial guess and applying a recurrence formula. Examples of iterative methods are Neumann series method, successive substitution method, fixed point method, etc.



  • Variational methods: These methods involve finding the solution that minimizes or maximizes a certain functional related to the integral equation. Examples of variational methods are Rayleigh-Ritz method, least squares method, etc.



  • Integral transform methods: These methods involve applying an integral transform (such as Fourier transform, Laplace transform, etc.) to both sides of the equation and solving for the transformed function. Then, applying the inverse transform to obtain the original function.



Why are linear integral equations important?




Linear integral equations have many applications in various fields of science and engineering, such as potential theory, fluid mechanics, electromagnetism, heat transfer, elasticity, quantum mechanics, etc. They can be used to model phenomena that involve distributed or boundary sources, such as heat conduction in a rod with heat sources along its length, electric potential in a region with charges on its boundary, fluid flow past an obstacle with vortices on its surface, etc.


Applications in science and engineering




Here are some examples of how linear integral equations can be used to model real-world problems:


  • Potential theory: The Dirichlet problem is to find a harmonic function (a function that satisfies Laplace's equation $\nabla^2 u=0$) in a given region that takes prescribed values on its boundary. This problem can be solved by using Green's function (a solution of Laplace's equation with a point source at x=t), which satisfies $$u(x)=\int_\partial D G(x,t)\frac\partial u\partial n(t)dS_t - \int_\partial D u(t)\frac\partial G\partial n(x,t)dS_t$$ where D is the region, $\partial D$ is its boundary, n is the outward normal vector on $\partial D$, and dSt is the surface element on $\partial D$. This is an example of a Fredholm equation of the second kind.



  • Fluid mechanics: The Blasius boundary layer problem is to find the velocity profile (the variation of velocity along the direction perpendicular to the flow) of a viscous fluid flowing past a flat plate. This problem can be solved by using similarity variables (variables that reduce the number of independent variables by exploiting symmetries or scaling properties), which transform the problem into an ordinary differential equation $$f''' + \frac12ff''=0$$ where f' denotes differentiation with respect to $\eta$, a similarity variable defined as $\eta=y\sqrt\fracU\nu x$, where y is the distance from the plate, U is the free stream velocity, $\nu$ is the kinematic viscosity, and x is the distance along the plate. This ordinary differential equation can be further transformed into an integral equation $$f(\eta)=\eta - \frac12\int_0^\eta f'(\xi)^2 d\xi$$ which can be solved by iterative methods. This is an example of a Volterra equation of the second kind.



  • Electromagnetism: The Helmholtz equation $(\nabla^2 + k^2)\phi=0$ describes the propagation of electromagnetic waves in free space or homogeneous media, where k is the wave number related to the frequency and wavelength of the wave. The solution $\phi$ represents either the electric potential or the magnetic potential, depending on the choice of gauge. The Helmholtz equation can be solved by using Green's function (a solution of Helmholtz equation with a point source at x=t), which satisfies $$\phi(x n)(x,t)dS_t$$ where V is the volume containing the source f(t), S is its surface, n is the outward normal vector on S, and dVt and dSt are the volume and surface elements. This is an example of a Fredholm equation of the first kind.



  • Heat transfer: The heat equation $\frac\partial u\partial t = \alpha \nabla^2 u + Q$ describes the temperature distribution u(x,t) in a medium with thermal diffusivity $\alpha$ and heat source Q(x,t). If Q(x,t) is independent of time, then the steady-state solution satisfies $\nabla^2 u + \fracQ\alpha = 0$, which is a Poisson equation. This equation can be solved by using Green's function (a solution of Poisson equation with a point source at x=t), which satisfies $$u(x) = -\frac14\pi\alpha\int_V G(x,t)Q(t)dV_t + \frac14\pi\int_S G(x,t)\frac\partial u\partial n(t)dS_t - \frac14\pi\int_S u(t)\frac\partial G\partial n(x,t)dS_t$$ where V is the volume containing the heat source Q(t), S is its surface, n is the outward normal vector on S, and dVt and dSt are the volume and surface elements. This is an example of a Fredholm equation of the second kind.



Advantages over differential equations




Linear integral equations have some advantages over differential equations in modeling certain problems. Some of these advantages are:


  • Integral equations can handle singularities or discontinuities more easily than differential equations, as they do not require differentiability of the solution or the coefficients.



  • Integral equations can incorporate boundary conditions more naturally than differential equations, as they do not require auxiliary conditions or compatibility conditions.



  • Integral equations can reduce the dimensionality of the problem by integrating over one or more variables, which can simplify the analysis and computation.



  • Integral equations can provide a global perspective of the problem by relating the values of the solution at different points, which can reveal some hidden symmetries or properties.



Who is Shanti Swarup and what is his book about?




Shanti Swarup was an Indian mathematician and scientist who made significant contributions to colloid chemistry and physical chemistry. He was also a pioneer of scientific research and education in India, and he founded several institutions and laboratories for promoting science and technology.


Biography and achievements




Shanti Swarup was born on 21 February 1894 in Bhera, Punjab region of British India (now in Pakistan). His father died when he was eight months old, and he was raised by his maternal grandfather, who was an engineer. He developed an interest in science and engineering from an early age, and he also showed a talent for poetry and drama. He completed his BSc in physics from Forman Christian College in Lahore in 1916, and his MSc in chemistry from University of Punjab in 1919. He then received a scholarship to study abroad, but he could not find a berth on English ships due to the First World War. He decided to go to London via Japan, but he was stranded in Japan for six months due to lack of funds. He finally reached London in 1920, where he joined University College London as a research student under Frederick G Donnan. He obtained his DSc in 1921 for his work on emulsions and colloids.


Shanti Swarup returned to India in 1921 and joined Banaras Hindu University as a professor of chemistry. He established a department of chemical engineering and a research laboratory for colloid chemistry. He also started a journal called Journal of Colloid Science (later renamed as Proceedings of the Indian Academy of Sciences). He worked on various topics such as surface tension, adsorption, catalysis, viscosity, light scattering, etc. He also collaborated with other scientists such as C V Raman, Meghnad Saha, K S Krishnan, etc. He published over 100 research papers and several books on colloid chemistry.


In 1939, Shanti Swarup was appointed as the first director-general of Council of Scientific and Industrial Research (CSIR), which was established by the government of India to promote scientific and industrial research. He played a key role in setting up various laboratories and institutes under CSIR, such as National Chemical Laboratory (NCL), National Physical Laboratory (NPL), Central Food Technological Research Institute (CFTRI), etc. He also initiated several schemes for funding research projects, awarding fellowships, organizing conferences, etc. He was instrumental in creating a scientific culture and infrastructure in India.


Shanti Swarup received many honors and awards for his contributions to science and society. He was elected as a Fellow of the Royal Society (FRS) in 1943. He was knighted by King George VI in 1941. He was awarded the Padma Bhushan by the government of India in 1954. He was also associated with various national and international organizations such as Indian National Science Academy (INSA), Indian Science Congress Association (ISCA), International Union of Pure and Applied Chemistry (IUPAC), etc.


Shanti Swarup died on 1 January 1955 in New Delhi at the age of 60. His legacy lives on through his scientific work, his institutions, his students, and his awards. In 1958, CSIR instituted the Shanti Swarup Bhatnagar Prize for Science and Technology to recognize outstanding contributions by Indian scientists under the age of 45.


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