Analytical and Computational Methods of Advanced Engineering Mathematics

1 Numerical Analysis.- 1.1 The Nature of Numerical Analysis.- 1.2 Polynomial Interpolation.- 1.3 Numerical Integration and Differentiation.- 1.4 Solution of Equations.- 1.5 Inverse Functions.- 1.6 Implicit Functions.- 1.7 Numerical Summation of Infinite Series.- 2 Ordinary Differential Equations of First Order.- 2.1 The Nature of Differential Equations.- 2.2 Separable Equations.- 2.3 Linear First-Order Equations.- 2.4 Exact Equations.- 2.5 Applications to Some Second-Order Equations.- 2.6 The Initial Value Problem.- 2.7 Numerical Methods for the Initial Value Problem.- 3 Ordinary Differential Equations of Higher Order.- 3.1 Examples from Engineering and Physics.- 3.2 Linear Second-Order Equations — Structure of Solutions.- 3.3 Linear Second-Order Equations with Constant Coefficients.- 3.4 Linear Second-Order Equations with Analytic Coefficients.- 3.5 Numerical Methods for Second-Order Equations.- 3.6 Linear Equations of Order n > 2.- 4 The Laplace Transform.- 4.1 The Nature of the Laplace Transform.- 4.2 The Laplace Transforms of Some Elementary Functions.- 4.3 Operational Rules for the Laplace Transform.- 4.4 Applications to Differential Equations.- 4.5 Applications to Systems of Differential Equations.- 5 Linear Algebra.- 5.1 Systems of Linear Equations.- 5.2 The Gauss Elimination Method.- 5.3 Vector Spaces.- 5.4 Matrices and Matrix Algebra.- 5.5 The Fundamental Theorem of Linear Algebra.- 5.6 Determinants and Cramer’s Rule.- 5.7 Eigenvalues and Eigenvectors.- 6 Vector Analysis.- 6.1 Vector Algebra.- 6.2 Vector Calculus of Curves in Space.- 6.3 Vector Calculus of Surfaces in Space.- 6.4 Calculus of Scalar and Vector Fields.- 6.5 Integral Theorems of Vector Calculus.- 6.6 X-Ray Diffraction and Crystal Structure.- 7 Partial Differential Equations of MathematicalPhysics.- 7.1 Vibrating Strings: D’Alembert’s Wave Equation.- 7.2 Heat Diffusion in Rods: Fourier’s Heat Equation.- 7.3 Heat Diffusion in Plates.- 7.4 Steady-State Heat Diffusion in Plates: The Laplace Equation.- 7.5 Vibrations of Drums.- 7.6 Heat Diffusion in Solids.- 7.7 Steady-State Heat Diffusion in Solids.- 8 Fourier Analysis and Sturm-Liouville Theory.- 1 Fourier Series.- II Fourier Integrals.- III Sturm-Liouville Theory.- 9 Boundary Value Problems of Mathematical Physics.- 9.1 Heat Diffusion in One Dimension.- 9.2 Vibration of Strings and Traveling Waves.- 9.3 Steady-State Diffusion of Heat in Plates.- 9.4 Transient Diffusion of Heat in Plates.- 9.5 Vibrations of Drums.- 9.6 Steady-State Diffusion of Heat in Solids.- 9.7 The Laplace Transform Method.- Appendix: Answers and Hints to Selected Exercises.- References.