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Partial Differential Equations for Scientists and Engineers

John Keats

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An using Fourier and Laplace Transforms

Partial differential equations (PDEs) are a powerful tool for modeling a wide variety of physical phenomena, from the flow of fluids to the propagation of waves. They are used in a wide range of scientific and engineering disciplines, including physics, engineering, mathematics, and biology.

Partial Differential Equations for Scientists and Engineers (Dover on Mathematics)

Partial Differential Equations for Scientists and Engineers (Dover Books on Mathematics)

by Stanley J. Farlow

4.5 out of 5

Language	:	English
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This book provides a comprehensive to PDEs for scientists and engineers. It covers a wide range of topics, including the separation of variables, the Fourier series, the Laplace transform, and the wave equation. It also provides numerous examples and exercises to help students understand the material.

What is a partial differential equation?

A partial differential equation is an equation that involves the partial derivatives of a function. Partial derivatives are derivatives with respect to one or more of the independent variables in the function. For example, the following equation is a partial differential equation in two variables:

∂u/∂x + ∂u/∂y = 0

This equation says that the partial derivative of u with respect to x plus the partial derivative of u with respect to y is equal to zero.

Why are partial differential equations important?

PDEs are important because they can be used to model a wide variety of physical phenomena. For example, the following PDE can be used to model the flow of heat in a solid:

∂u/∂t = k∇^2u

In this equation, u is the temperature of the solid, t is time, and k is the thermal conductivity of the solid. The Laplacian operator, ∇^2, is a second-order partial differential operator that measures the curvature of a function.

How are partial differential equations solved?

There are a variety of methods for solving PDEs. Some of the most common methods include the separation of variables, the Fourier series, the Laplace transform, and the method of characteristics. The method of separation of variables involves finding a solution to the PDE that can be written as a product of functions of each of the independent variables. The Fourier series involves expanding the solution to the PDE as a series of trigonometric functions. The Laplace transform involves transforming the PDE into a simpler equation in the Laplace domain. The method of characteristics involves finding a set of curves along which the solution to the PDE is constant.

What are some applications of partial differential equations?

PDEs are used in a wide range of scientific and engineering disciplines, including physics, engineering, mathematics, and biology. Some of the most common applications of PDEs include:

Modeling the flow of fluids
Modeling the propagation of waves
Solving heat transfer problems
Solving elasticity problems
Solving electromagnetism problems

Summary

Partial differential equations are a powerful tool for modeling a wide variety of physical phenomena. They are used in a wide range of scientific and engineering disciplines, including physics, engineering, mathematics, and biology. This book provides a comprehensive to PDEs for scientists and engineers. It covers a wide range of topics, including the separation of variables, the Fourier series, the Laplace transform, and the wave equation. It also provides numerous examples and exercises to help students understand the material.

Partial Differential Equations for Scientists and Engineers (Dover Books on Mathematics)

by Stanley J. Farlow

4.5 out of 5