Advanced Partial Differential Equations

Applied Partial Differential Equations

Applied Partial Differential Equations

By

Sameer Kulkarni

Applied Partial Differential Equations

Sameer Kulkarni

ISBN - 9789361526404

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Preface

This book is designed to serve as a valuable resource for students, researchers, and professionals seeking a deeper understanding of the mathematical tools and techniques essential for tackling real-world problems.

Objective:

The primary aim of this book is to bridge the gap between theory and application, offering readers a balanced and insightful journey through the realm of PDEs. Whether you are a student embarking on a mathematical exploration or a seasoned practitioner seeking advanced insights, this book is crafted to provide a meaningful and accessible learning experience.

Key Features:

Foundational Concepts: The book begins with an introduction to fundamental concepts, definitions, and classifications of partial differential equations, providing a solid foundation for subsequent chapters.

In-depth Exploration: Each type of PDE—elliptic, parabolic, and hyperbolic—is explored in detail, with dedicated chapters elucidating their characteristics, solutions, and applications.

Numerical Methods: A comprehensive discussion on numerical methods equips readers with the skills to solve PDEs computationally, addressing challenges in accuracy and efficiency.

Interdisciplinary Applications: Practical applications in science and engineering are woven throughout the text, demonstrating the versatility and relevance of PDEs in fields such as fluid dynamics, heat conduction, and electromagnetism.

Advanced Topics: The book delves into advanced topics, including nonlinear PDEs, asymptotic methods, and multi-dimensional problems, providing a bridge to cutting-edge research and applications.

Case Studies: Real-world case studies and practical examples offer readers the opportunity to apply theoretical knowledge to authentic problem-solving scenarios.

Who Should Read:

This book is intended for undergraduate and graduate students studying mathematics, physics, engineering, or related disciplines, as well as professionals and researchers seeking a comprehensive reference in applied partial differential equations.

How to Use This Book:

Each chapter is designed for sequential learning, allowing readers to build a strong foundation before tackling more advanced topics. However, the book is also structured to facilitate selective reading for those interested in specific applications or methodologies.

Table of Contents

1. Introduction to Partial Differential Equations1

1.1 Definition and Classification1

1.2 Examples and Applications2

1.3 Basic Concepts and Notation5

2. First-Order Partial Differential Equations10

2.1 Characteristics and Canonical Forms10

2.2 Methods of Solution11

2.3 Conservation Laws12

3. Second-Order Elliptic Equations16

3.1 Laplace’s Equation16

3.2 Boundary Value Problems17

3.3 Fourier Series and Eigenvalue Problems17

4. Second-Order Parabolic Equations21

4.1 Heat Equation21

4.2 Initial and Boundary Value Problems22

4.3 Separation of Variables22

5. Second-Order Hyperbolic Equations27

5.1 Wave Equation27

5.2 Characteristics and Solutions28

5.3 Shock Waves28

6. Numerical Methods for Partial Differential Equations33

6.1 Finite Difference Methods33

6.2 Finite Element Methods33

6.3 Computational Techniques34

7. Advanced Topics in Applied Partial Differential Equations36

7.1 Nonlinear Partial Differential Equations36

7.2 Asymptotic Methods37

7.3 Multi-dimensional Problems38

8. Applications in Science and Engineering43

8.1 Fluid Dynamics43

8.2 Heat Conduction45

8.3 Electromagnetic Fields46

9. Green's Functions51

9.1 Introduction51

9.2 Properties51

9.3 Derivation and Physical Interpretation52

9.4 Applications in Boundary Value Problems52

9.5 Green's Functions for the Heat Equation, Wave Equation, and Laplace's Equation53

10. Asymptotic Methods59

10.1 Introduction59

10.2 Perturbation Methods59

10.3 Asymptotic Expansions60

10.4 WKB Approximation61

10.5 Applications in Boundary Layer Theory, Shock Waves, and Singular Perturbations63

11. Multi-dimensional Problems68

11.1 Introduction68

11.2 Applications in Fluid Dynamics, Solid Mechanics, and Electromagnetism69

11.3 Solutions in Cartesian, Cylindrical, and Spherical Coordinates71

11.4 Applications in Heat Transfer, Wave Propagation, and Electrostatics74

12. Applications of Partial Differential Equations80

12.1 Heat Transfer80

12.2 Fluid Dynamics81

12.3 Solid Mechanics81

12.4 Electromagnetism82

12.5 Quantum Mechanics83

12.6 Relativity Theory84

12.7 Applications in Engineering, Physics, and Mathematics85

13. Applications in Engineering92

13.1 Fluid Dynamics and Aerodynamics92

13.2 Heat Transfer and Thermodynamics96

13.3 Electromagnetics and Wave Propagation98

13.4 Structural Mechanics and Solid Dynamics102

13.5 Geophysics and Environmental Engineering104

14. Applications in Physics110

14.1 Quantum Mechanics and Wave Functions110

14.2 Relativity Theory and Gravitational Waves112

14.3 Electrodynamics and Quantum Field Theory113

14.4 Statistical Mechanics and Kinetic Theory115

15. Applications in Biology and Medicine121

15.1 Mathematical Modeling of Biological Systems121

15.2 Diffusion and Reaction-Diffusion Equations123

15.3 Biomechanics and Medical Imaging125

16. Applications in Finance and Economics131

16.1 Black-Scholes Equation and Option Pricing131

16.2 Financial Mathematics and Risk Analysis133

16.3 Economic Modeling and Game Theory136

Glossary142

Index148

CHAPTER 1 Introduction to Partial Differential Equations

1.1 Definition and Classification

Partial Differential Equations (PDEs) are mathematical equations that involve multiple independent variables and their partial derivatives. Unlike ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives, PDEs involve functions of multiple variables and their partial derivatives with respect to those variables.

A general form of a partial differential equation is often written as:

Fig 1.1 Partial Differential Equations

https://cdn1.byjus.com/wp-content/uploads/2019/07/PDE-Notation.png

Here,

u represents the unknown function of the variables x1,x2,…,xn and the equation relates the function and its partial derivatives. The specific form of F and the variables involved depend on the physical or mathematical problem being modelled.

PDEs are widely used in various scientific disciplines to describe phenomena such as heat conduction, fluid dynamics, electromagnetic fields, quantum mechanics, and many others. They provide a powerful tool for understanding and predicting the behaviour of systems with multiple interacting variables.

PDEs can be classified into different types based on their order, linearity, and the number of independent variables. The most common classifications include:

Order: The order of a PDE is determined by the highest order of the partial derivatives involved. For example, a second-order PDE involves second-order partial derivatives.

Linearity: PDEs can be linear or nonlinear. Linear PDEs satisfy the principle of superposition, meaning that if

1u1 and 2u2 are solutions, then

1+2au1+bu2 is also a solution, where a and b are constants.

Number of Variables: Depending on the number of independent variables, PDEs can be classified as partial differential equations (PDEs) or ordinary differential equations (ODEs).

Two fundamental types of PDEs are elliptic, parabolic, and hyperbolic, based on the nature of their solutions and the characteristics of the equations.

Elliptic PDEs:

These equations describe steady-state problems and have solutions that exhibit smooth behaviour. Examples include Laplace's equation and Poisson's equation.

Parabolic PDEs:

These equations describe phenomena that evolve over time and involve a balance between diffusion and convection. The heat equation is a classic example of a parabolic PDE.

Hyperbolic PDEs:

These equations describe wave-like behaviour and involve characteristics that determine the propagation of disturbances. Examples include the wave equation and the telegraph equation.

Solving PDEs analytically can be challenging, and often numerical methods, such as finite difference methods, finite element methods, and others, are employed to approximate solutions. The study of PDEs is crucial in understanding complex physical phenomena and plays a significant role in various scientific and engineering applications.

Classification

Partial Differential Equations (PDEs) can be classified based on various characteristics such as their order, linearity, and the number of independent variables. Additionally, PDEs can be categorised into different types depending on the nature of their solutions and the physical phenomena they model. Here are some common classifications:

Order of the PDE:

First-order PDEs:

Involves only first-order partial derivatives.

Second-order PDEs:

Involves second-order partial derivatives and is often encountered in physics and engineering.

Number of Variables:

Single-variable PDEs:

Involves a single independent variable and its partial derivatives.

Multivariable PDEs:

Involves more than one independent variable and their partial derivatives.

Linearity:

Linear PDEs: Satisfy the principle of superposition, meaning that if

1u1 and 2u2

are solutions, then

1+2au1 +bu2 is also a solution, where a and b are constants.

Nonlinear PDEs: Do not satisfy the principle of superposition and can exhibit more complex behaviour.

Homogeneous and Inhomogeneous:

Homogeneous PDEs:

Have zero on the right-hand side (RHS) of the equation.

Inhomogeneous PDEs:

Have a non-zero term on the RHS.

Classification Based on Physical Behavior:

Elliptic PDEs:

Involve steady-state problems and have solutions that exhibit smooth behaviour. Examples include Laplace's equation and Poisson's equation.

Parabolic PDEs:

Describe phenomena that evolve over time and involve a balance between diffusion and convection. Examples include the heat equation.

Hyperbolic PDEs:

Describe wave-like behaviour and involve characteristics that determine the propagation of disturbances. Examples include the wave equation and the telegraph equation.

Quasilinear and Fully Nonlinear:

Quasilinear PDEs:

The coefficients in the PDE depend on the solution itself but not on higher-order derivatives of the solution.

Fully nonlinear PDEs:

The coefficients in the PDE depend on both the solution and its higher-order derivatives.

Understanding the classification of PDEs is crucial for selecting appropriate solution methods, as different types of equations may require distinct techniques for analysis and numerical solution. Depending on the specific problem at hand, researchers and engineers choose the appropriate approach and tools to tackle PDEs and gain insights into the behaviour of physical systems.

1.2 Examples and Applications

Partial Differential Equations (PDEs) arise in various scientific and engineering disciplines to describe the behaviour of physical phenomena involving multiple variables. Here are some examples of commonly encountered PDEs:

Heat Equation (Parabolic PDE):

This equation describes the distribution of heat in a conducting material over time, where

u is the temperature,

t is time, and

α is the thermal diffusivity.

Wave Equation (Hyperbolic PDE):

The wave equation models the propagation of waves, such as sound waves or electromagnetic waves, where

u represents the wave amplitude,

t is time, and

c is the wave speed.

Laplace's Equation (Elliptic PDE):

Laplace's equation describes steady-state situations, such as the electrostatic potential in the absence of charges.

u is the scalar field.

Poisson's Equation (Elliptic PDE):

∇2u=f

Poisson's equation is similar to Laplace's equation but includes a source term

f, making it applicable to situations with distributed sources.

Fig 1.2 Classification of PDE’s

https://slideplayer.com/slide/4693769/15/images/7/Advection+Equation+The+method+we+study+used+to+solve+this+equation+can+be+generalized%253A+-+vectors+u%2528x%252Cy%252Cz%252Ct%2529.jpg

Advection Equation (First-Order Linear PDE):

Fig 1.3 Advection Equation

https://slideplayer.com/slide/4693769/15/images/7/Advection+Equation+The+method+we+study+used+to+solve+this+equation+can+be+generalized%253A+-+vectors+u%2528x%252Cy%252Cz%252Ct%2529.jpg

Burgers' Equation (Quasilinear PDE):

Fig 1.4 Burger’s Equation

https://slideplayer.com/slide/17348538/101/images/70/Quasi-linear+1st+order+PDEs+Formation+of+shocks+%26+expansion+wave.jpg

Burgers' equation combines advection and diffusion and is used to model phenomena like shock waves in fluid dynamics.

Maxwell's Equations (System of PDEs):

Maxwell's equations describe the behaviour of electric and magnetic fields in electrodynamics. They form a system of four coupled PDEs.

Fig 1.5 Maxwell’s Equation

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UgAAAW4AAACKCAMAAAC93lC

Navier-Stokes Equations (System of PDEs):

The Navier-Stokes equations describe the motion of viscous fluid and are fundamental in fluid dynamics.

Fig 1.6 Navier Stokes Equation

https://nuclear-power.com/wp-content/uploads/2017/11/Navier-Stokes-Equations-definition.png

These examples represent just a small subset of the diverse range of PDEs encountered in science and engineering. The specific form of the equation depends on the physical context and the phenomena being modelled. Analytical and numerical methods are employed to solve these equations and gain insights into the behaviour of complex systems.

1.3 Basic Concepts and Notation

Understanding partial differential equations (PDEs) involves grasping several fundamental concepts. Here are some basic concepts related to PDEs:

Partial Derivatives:

PDEs involve partial derivatives, which indicate how a function changes concerning each of its independent variables. For example,∂x/∂u represents the partial derivative of u with respect to x.

Dependent and Independent Variables:

The unknown function in a PDE is typically denoted as u(x1,x2,…,xn,t), where u is the dependent variable, and x1,x2,…,xn,t are the independent variables.

Order of a PDE:

The order of a PDE is determined by the highest order of the partial derivatives involved. For example,

Linear and Nonlinear PDEs:

PDEs are classified as linear or nonlinear based on whether they satisfy the principle of superposition. Linear PDEs allow for the formation of linear combinations of solutions, while nonlinear PDEs do not.

Initial and Boundary Conditions:

To solve a PDE, initial conditions (for time-dependent problems) and boundary conditions must be specified. These conditions provide additional information needed to determine the unique solution.

Homogeneous and Inhomogeneous PDEs:

Homogeneous PDEs have zero on the right-hand side, whereas inhomogeneous PDEs have a non-zero term on the right-hand side.

Elliptic, Parabolic, and Hyperbolic Equations:

PDEs are often categorised into three types based on their characteristics and behaviour:

Elliptic PDEs: Steady-state problems, such as Laplace's equation.

Parabolic PDEs: Describe phenomena evolving over time, like the heat equation.

Hyperbolic PDEs: Involve wave-like behaviour, for example, the wave equation.

Boundary Value Problems (BVP) and Initial Value Problems (IVP):

A boundary value problem involves specifying values of the unknown function on the boundary of the domain, while an initial value problem involves specifying values at a particular initial time.

Conservation Laws:

Some PDEs arise from physical principles and represent conservation laws. For example, the continuity equation in fluid dynamics expresses the conservation of mass.

Method of Characteristics:

This method is often employed to solve first-order PDEs. It involves finding characteristic curves along which the PDE reduces to an ordinary differential equation (ODE).

Analytical and Numerical Methods:

Analytical methods, such as separation of variables, Fourier transforms, and Green's functions, are used for obtaining exact solutions when possible. Numerical methods, like finite difference and finite element methods, are applied for approximate solutions.

Existence and Uniqueness of Solutions:

Ensuring the existence and uniqueness of solutions is crucial in the study of PDEs. The well-posedness of a problem depends on the formulation of the PDE and the provided conditions.

These basic concepts provide a foundation for studying and solving partial differential equations, which are essential in modelling various physical phenomena in science and engineering.

Quick Recap

PDEs can be classified into different types based on their order, linearity, and the number of independent variables. The most common classifications include:

Order: The order of a PDE is determined by the highest order of the partial derivatives involved. For example, a second-order PDE involves second-order partial derivatives.

Linearity: PDEs can be linear or nonlinear. Linear PDEs satisfy the principle of superposition, meaning that if

1u1 and 2u2 are solutions, then

1+2au1+bu2 is also a solution, where a and b are constants.

Number of Variables: Depending on the number of independent variables, PDEs can be classified as partial differential equations (PDEs) or ordinary differential equations (ODEs).

Two fundamental types of PDEs are elliptic, parabolic, and hyperbolic, based on the nature of their solutions and the characteristics of the equations.

•Elliptic PDEs: These equations describe steady-state problems and have solutions that exhibit smooth behaviour. Examples include Laplace's equation and Poisson's equation.

•Parabolic PDEs: These equations describe phenomena that evolve over time and involve a balance between diffusion and convection. The heat equation is a classic example of a parabolic PDE.

•Hyperbolic PDEs: These equations describe wave-like behaviour and involve characteristics that determine the propagation of disturbances. Examples include the wave equation and the telegraph equation.

Solving PDEs analytically can be challenging, and often numerical methods, such as finite difference methods, finite element methods, and others, are employed to approximate solutions. The study of PDEs is crucial in understanding complex physical phenomena and plays a significant role in various scientific and engineering applications.

Partial Differential Equations (PDEs) can be classified based on various characteristics such as their order, linearity, and the number of independent variables. Additionally, PDEs can be categorised into different types depending on the nature of their solutions and the physical phenomena they model. Here are some common classifications:

Order of the PDE:

•First-order PDEs: Involves only first-order partial derivatives.

•Second-order PDEs: Involves second-order partial derivatives and is often encountered in physics and engineering.

Number of Variables:

•Single-variable PDEs: Involves a single independent variable and its partial derivatives.

•Multivariable PDEs: Involves more than one independent variable and their partial derivatives.

Linearity:

•Linear PDEs: Satisfy the principle of superposition, meaning that if

•1u1 and 2u2

•are solutions, then

•1+2au1 +bu2 is also a solution, where a and b are constants.

•Nonlinear PDEs: Do not satisfy the principle of superposition and can exhibit more complex behaviour.

Homogeneous and Inhomogeneous:

•Homogeneous PDEs: Have zero on the right-hand side (RHS) of the equation.

•Inhomogeneous PDEs: Have a non-zero term on the RHS.

Question Bank

1.What is the fundamental difference between partial differential equations (PDEs) and ordinary differential equations (ODEs)?

2.Explain the meaning of the dependent and independent variables in the context of PDEs.

3.Define the order of a partial differential equation and provide an example of a second-order PDE.

4.Discuss the linearity of PDEs and how it influences the solution space.

5.Differentiate between homogeneous and inhomogeneous PDEs, providing an example of each.

6.Explain the significance of initial conditions and boundary conditions in solving PDEs.

7.Classify PDEs as elliptic, parabolic, or hyperbolic and provide an example of each type.

8.What role do conservation laws play in the formulation of certain PDEs? Provide an example.

9.Briefly describe the method of characteristics and its application in solving PDEs.

10.Discuss the concept of well-posed problems in the context of PDEs.

11.Explain the finite difference method and how it can be applied to solve a simple PDE.

12.Discuss the advantages and limitations of finite element methods in solving partial differential equations.

13.Compare and contrast numerical and analytical methods for solving PDEs.

14.Illustrate the use of numerical simulations in solving time-dependent PDEs with a practical example.

15.Discuss the stability and convergence criteria for numerical solutions of PDEs.

16.Explore the concept of weak solutions in the context of PDEs and their importance in mathematical analysis.

17.Discuss the concept of characteristic surfaces in hyperbolic PDEs and their role in determining the solution behavior.

18.Investigate the use of Green's functions in solving inhomogeneous PDEs and provide an example.

19.Explain the concept of Sobolev spaces and their relevance in the study of PDEs.

20.Discuss the concept of uniqueness and existence of solutions for nonlinear PDEs and the challenges associated with them.

21.Model the diffusion of a substance in a two-dimensional space using a diffusion equation and solve for a given initial distribution.

22.Formulate and solve a PDE to describe the steady-state temperature distribution in a thin metal plate subject to specific boundary conditions.

23.Discuss the application of PDEs in image processing, specifically in image denoising or edge detection.

24.Explore how PDEs are employed in financial mathematics, for instance, in option pricing models.

25.Model and solve a PDE to describe the propagation of a disease in a population, considering spatial and temporal factors.

26.Implement a finite element method to solve a two-dimensional elliptic PDE and discuss the steps involved in the numerical solution.

27.Compare the advantages and disadvantages of finite volume methods with finite difference methods in solving hyperbolic PDEs.

28.Discuss the role of adaptive mesh refinement in numerical solutions of PDEs and its impact on computational efficiency.

29.Explore the use of high-performance computing (HPC) in solving large-scale PDE