Solving Nonlinear Partial Differential Equations with Maple and Mathematica [electronic resource] /
Shingareva, Inna.author.
Lizárraga-Celaya, Carlos.author.
The emphasis of this work is on constructing different types of solutions (exact, approximate analytical, numerical, graphical) of numerous nonlinear PDEs correctly, easily, and quickly. The reader can learn a wide variety of techniques and solve numerous nonlinear PDEs included and many other differential equations, simplifying and transforming the equations and solutions, arbitrary functions and parameters, presented in the book). Numerous comparisons and relationships between various types of solutions, different methods and approaches are provided, the results obtained in Maple and Mathematica, facilitates a deeper understanding of the subject.
Description based upon print version of record.
Solving Nonlinear Partial Differential Equationswith Maple and Mathematica; Preface; Contents; Chapter 1 Introduction; 1.1 Basic Concepts; 1.1.1 Types of Partial Differential Equations; 1.1.2 Nonlinear PDEs and Systems Arising in Applied Sciences; 1.1.3 Types of Solutions of Nonlinear PDEs; 1.2 Embedded Analytical Methods; 1.2.1 Nonlinear PDEs; 1.2.2 Nonlinear PDEs with Initial and/or Boundary Conditions; 1.2.3 Nonlinear Systems; 1.2.4 Nonlinear Systems with Initial and/or Boundary Conditions; Chapter 2 Algebraic Approach; 2.1 Point Transformations
2.1.1 Transformations of Independent and/or Dependent Variables2.1.2 Hodograph Transformation; 2.2 Contact Transformations; 2.2.1 Legendre Transformation; 2.2.2 Euler Transformation; 2.3 Transformations Relating Differential Equations; 2.3.1 B ̈acklund Transformations; 2.3.2 Miura Transformation; 2.3.3 Gardner Transformation; 2.4 Linearizing and Bilinearizing Transformations; 2.4.1 Hopf-Cole Transformation; 2.4.2 Hopf-Cole-type Transformation; 2.5 Reductions of Nonlinear PDEs; 2.5.1 Traveling Wave Reductions; 2.5.2 Ansatz Methods; 2.5.3 Self-Similar Reductions; 2.6 Separation of Variables
2.6.1 Ordinary Separation of Variables2.6.2 Partial Separation of Variables; 2.6.3 Generalized Separation of Variables; 2.6.4 Functional Separation of Variables; 2.7 Transformation Groups; 2.7.1 One-Parameter Groups of Transformations; 2.7.2 Group Analysis; 2.7.3 Invariant Solutions; 2.8 Nonlinear Systems; 2.8.1 Traveling Wave Reductions; 2.8.2 Special Reductions; 2.8.3 Separation of Variables; Chapter 3 Geometric-Qualitative Approach; 3.1 Method of Characteristics; 3.1.1 Characteristic Directions. General Solution; 3.1.2 Integral Surfaces. Cauchy Problem; 3.1.3 Solution Profile at Infinity
3.2 Generalized Method of Characteristics3.2.1 Complete Integrals. General Solution; 3.2.2 The Monge Cone. Characteristic Directions; 3.2.3 Integral Surfaces. Cauchy Problem; 3.3 Qualitative Analysis; 3.3.1 Nonlinear PDEs; 3.3.2 Nonlinear Systems; Chapter 4 General Analytical Approach. Integrability; 4.1 Painlevé Test and Integrability; 4.1.1 Painlev ́e Property and Test; 4.1.2 Truncated expansions; 4.2 Complete Integrability. Evolution Equations; 4.2.1 Conservation Laws; 4.2.2 Nonlinear Superposition Formulas; 4.2.3 Hirota Method; 4.2.4 Lax Pairs; 4.2.5 Variational Principle
4.3 Nonlinear Systems. Integrability ConditionsChapter 5 Approximate Analytical Approach; 5.1 Adomian Decomposition Method; 5.1.1 Adomian Polynomials; 5.1.2 Nonlinear PDEs; 5.1.3 Nonlinear Systems; 5.2 Asymptotic Expansions. Perturbation Methods; 5.2.1 Nonlinear PDEs; 5.2.2 Nonlinear Systems; Chapter 6 Numerical Approach; 6.1 Embedded Numerical Methods; 6.1.1 Nonlinear PDEs; 6.1.2 Specifying Classical Numerical Methods; 6.1.3 Nonlinear Systems; 6.2 Finite Difference Methods; 6.2.1 Evolution Equations; 6.2.2 Interaction of Solitons; 6.2.3 Elliptic Equations
Chapter 7 Analytical-Numerical Approach
Includes bibliographical references.
The emphasis of this work is on constructing different types of solutions (exact, approximate analytical, numerical, graphical) of numerous nonlinear PDEs correctly, easily, and quickly. The reader can learn a wide variety of techniques and solve numerous nonlinear PDEs included and many other differential equations, simplifying and transforming the equations and solutions, arbitrary functions and parameters, presented in the book). Numerous comparisons and relationships between various types of solutions, different methods and approaches are provided, the results obtained in Maple and Mathematica, facilitates a deeper understanding of the subject.
Maple (Computer file)
Mathematica (Computer file)
Partial differential equations.
Statistical physics.
Applied mathematics.
Engineering mathematics.
Partial Differential Equations.
Applications of Nonlinear Dynamics and Chaos Theory.
Mathematical and Computational Engineering.