Multiplicative Partial Differential Equations

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Half Title
Series Page
Title Page
Copyright Page
Contents
Preface
1. General Introduction
2. Classification of Second-Order Multiplicative Partial Differential Equations
3. Classification and Canonical Forms
	3.1. Multiplicative Linear MPDEs
		3.1.1. The multiplicative hyperbolic case
		3.1.2. The multiplicative elliptic case
		3.1.3. The parabolic case
	3.2. Classification and Canonical Form of Multiplicative Quasilinear MPDE
		3.2.1. The multiplicative hyperbolic case
		3.2.2. The multiplicative elliptic case
		3.2.3. The multiplicative parabolic case
	3.3. n Multiplicative Independent Variables
	3.4. Classification of First-Order Systems with Two Multiplicative Independent Variables
	3.5. Advanced Practical Exercises
4. The Multiplicative Wave Equation
	4.1. The One-Dimensional Multiplicative Wave Equation
		4.1.1. The Cauchy problem and D\'Alambert formula
		4.1.2. The Cauchy problem for the nonhomogeneous wave equation
		4.1.3. Separation of multiplicative variables
		4.1.4. The energy method: uniqueness
	4.2. The Multiplicative Wave Equation in R3*
		4.2.1. Radially symmetric solutions
		4.2.2. The Cauchy problem
	4.3. The Two-Dimensional Wave Equation
	4.4. The (2n + 1)-Dimensional Wave Equation
	4.5. The (2n) − 1-Dimensional Wave Equation
	4.6. The Cauchy Problem for a Nonlinear Hyperbolic Equation
	4.7. The Riemann Function
	4.8. Advanced Practical Exercises
5. The Heat Equation
	5.1. The Weak Maximum Principle
	5.2. The Strong Maximum Principle
	5.3. The Cauchy Problem
		5.3.1. The fundamental solution
		5.3.2. The Cauchy problem
	5.4. The Mean Value Formula
	5.5. The Maximum Principle for the Cauchy Problem
	5.6. The Method of Separation of Variables
	5.7. The Energy Method: Uniqueness
	5.8. Advanced Practical Exercises
6. The Laplace Equation
	6.1. Basic Properties of Elliptic Problems
	6.2. The Fundamental Solution
	6.3. Integral Representation of Harmonic Functions
	6.4. Mean Value Formulas
	6.5. Strong Maximum Principle: Uniqueness
	6.6. The Poisson Equation
	6.7. The Green Function of the Dirichlet Problem
	6.8. The Poisson Formula for a Multiplicative Ball
	6.9. Theorems of Liouville and Harnack
	6.10. Separation of Variables
		6.10.1. Multiplicative rectangles
		6.10.2. Multiplicative circular domains
	6.11. Advanced Practical Exercises
7. The Cauchy‒Kovalevskaya Theorem
	7.1. Analytic Functions of One Variable
	7.2. Autonomous Multiplicative Differential Equations
	7.3. Systems of Ordinary Differential Equations
	7.4. Partial Differential Equations
References
Index