## Methods of Applied Mathematics for Engineers and ScientistsBased on course notes from over twenty years of teaching engineering and physical sciences at Michigan Technological University, Tomas Co's engineering mathematics textbook is rich with examples, applications and exercises. Professor Co uses analytical approaches to solve smaller problems to provide mathematical insight and understanding, and numerical methods for large and complex problems. The book emphasises applying matrices with strong attention to matrix structure and computational issues such as sparsity and efficiency. Chapters on vector calculus and integral theorems are used to build coordinate-free physical models with special emphasis on orthogonal co-ordinates. Chapters on ODEs and PDEs cover both analytical and numerical approaches. Topics on analytical solutions include similarity transform methods, direct formulas for series solutions, bifurcation analysis, Lagrange–Charpit formulas, shocks/rarefaction and others. Topics on numerical methods include stability analysis, DAEs, high-order finite-difference formulas, Delaunay meshes, and others. MATLABŪ implementations of the methods and concepts are fully integrated. |

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### Contents

Solution of Multiple Equations | 54 |

Matrix Analysis | 99 |

VECTORS AND TENSORS | 147 |

Vector Integral Theorems | 204 |

ORDINARY DIFFERENTIAL EQUATIONS | 233 |

Additional Details and Fortiﬁcation for Chapter 6 700 | 235 |

Numerical Solution of Initial and Boundary Value Problems | 273 |

Additional Details and Fortiﬁcation for Chapter 7 715 | 303 |

Additional Details and Fortiﬁcation for Chapter 10 771 | 379 |

Linear Partial Differential Equations | 405 |

Integral Transform Methods | 450 |

L3 Dirichlet Conditions and the Fourier Integral Theorem 819 | 456 |

Finite Difference Methods | 483 |

Method of Finite Elements | 523 |

Additional Details and Fortiﬁcation for Chapter 14 867 | 541 |

561 | |

Series Solutions of Linear Ordinary Differential Equations | 347 |

Additional Details and Fortiﬁcation for Chapter 9 745 | 354 |

PARTIAL DIFFERENTIAL EQUATIONS | 377 |

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apply approach arbitrary constants Bessel functions boundary conditions characteristic coefﬁcients column Consider curves deﬁned derivatives determine diagonal difference equation differential equation given Dirichlet discussed divergence theorem domain eigenvalues eigenvectors equilibrium points error Euler method evaluate exact solution EXAMPLE explicit ﬁnd ﬁnite difference approximation ﬁnite element method ﬁrst ﬁxed ﬂow ﬂux formulas Fourier transform gradient homogeneous implicit independent variables initial conditions inverse known Laplace transform Legendre limit cycle line integral Lyapunov Lyapunov function Lyapunov stable MATLAB Neumann nodes nonhomogeneous nonlinear nonsingular obtain ordinary differential equations orthogonal parameters partial differential equation path plot polynomials properties radius region respectively Runge-Kutta methods satisﬁes scalar scheme second-order series solution shown in Figure similarity transformation singular solution is given solve speciﬁc stability sufﬁcient surface integral tensor trajectories triangular value problems vector ﬁeld yield yk+1 zero