Numerical Methods in Engineering & Science
In: Springer eBook Collection
1 Introduction -- 1.1 What are numerical methods? -- 1.2 Numerical methods versus numerical analysis -- 1.3 Why use numerical methods? -- 1.4 Approximate equations and approximate solutions -- 1.5 The use of numerical methods -- 1.6 Errors -- 1.7 Non-dimensional equations -- 1.8 The use of computers -- 2 The solution of equations -- 2.1 Introduction -- 2.2 Location of initial estimates -- 2.3 Interval halving -- 2.4 Simple iteration -- 2.5 Convergence -- 2.6 Aitken's extrapolation -- 2.7 Damped simple iteration -- 2.8 Newton-Raphson method -- 2.9 Extended Newton's method -- 2.10 Other iterative methods -- 2.11 Polynomial equations -- 2.12 Bairstow's method 56 Worked examples 58 Problems -- 3 Simultaneous equations -- 3.1 Introduction -- 3.2 Elimination methods -- 3.3 Gaussian elimination -- 3.4 Extensions to the basic algorithm -- 3.5 Operation count for the basic algorithm -- 3.6 Tridiagonal systems -- 3.7 Extensions to the Thomas algorithm -- 3.8 Iterative methods for linear systems -- 3.9 Matrix inversion -- 3.10 The method of least squares -- 3.11 The method of differential correction -- 3.12 Simple iteration for non-linear systems -- 3.13 Newton's method for non-linear systems -- Worked examples -- Problems -- 4 Interpolation, differentiation and integration -- 4.1 Introduction -- 4.2 Finite difference operators -- 4.3 Difference tables -- 4.4 Interpolation -- 4.5 Newton's forward formula -- 4.6 Newton's backward formula -- 4.7 Stirling's central difference formula -- 4.8 Numerical differentiation -- 4.9 Truncation errors -- 4.10 Summary of differentiation formulae -- 4.11 Differentiation at non-tabular points: maxima and minima -- 4.12 Numerical integration -- 4.13 Error estimation -- 4.14 Integration using backward differences -- 4.15 Summary of integration formulae -- 4.16 Reducing the truncation error 146 Worked examples 149 Problems -- 5 Ordinary differential equations -- 5.1 Introduction -- 5.2 Euler's method -- 5.3 Solution using Taylor's series -- 5.4 The modified Euler method -- 5.5 Predictor-corrector methods -- 5.6 Milne's method, Adams' method, and Hamming's method -- 5.7 Starting procedure for predictor-corrector methods -- 5.8 Estimation of error of predictor-corrector methods -- 5.9 Runge-Kutta methods -- 5.10 Runge-Kutta-Merson method -- 5.11 Application to higher-order equations and to systems -- 5.12 Two-point boundary value problems -- 5.13 Non-linear two-point boundary value problems 198 Worked examples 199 Problems -- 6 Partial differential equations I — elliptic equations -- 6.1 Introduction -- 6.2 The approximation of elliptic equations -- 6.3 Boundary conditions -- 6.4 Non-dimensional equations again -- 6.5 Method of solution -- 6.6 The accuracy of the solution -- 6.7 Use of Richardson's extrapolation -- 6.8 Other boundary conditions -- 6.9 Relaxation by hand-calculation -- 6.10 Non-rectangular solution regions -- 6.11 Higher-order equations 238 Problems -- 7 Partial differential equations II — parabolic equations -- 7.1 Introduction -- 7.2 The conduction equation -- 7.3 Non-dimensional equations yet again -- 7.4 Notation -- 7.5 An explicit method -- 7.6 Consistency -- 7.7 The Dufort-Frankel method -- 7.8 Convergence -- 7.9 Stability -- 7.10 An unstable finite difference approximation -- 7.11 Richardson's extrapolation 261 Worked examples 262 Problems -- 8 Integral methods for the solution of boundary value problems -- 8.1 Introduction -- 8.2 Integral methods -- 8.3 Implementation of integral methods 271 Worked examples 278 Problems -- Suggestions for further reading.
