Numerical Integration and Differentiation

Numerical Integration and Differentiation

Differentiation Formulas 655 23.2 Richardson Extrapolation 658 23.3 Derivatives of Unequally Spaced Data 660 23.4 Derivatives and Integrals for Data with Errors 661 23.5 Partial Derivatives 662 23.6 Numerical Integration/Differentiation with Software Packages 663 Problems 670

CHAPTER 24 Case Studies: Numerical Integration and Differentiation 673

24.1 Integration to Determine the Total Quantity of Heat (Chemical/Bio Engineering) 673

24.2 Effective Force on the Mast of a Racing Sailboat (Civil/Environmental Engineering) 675

24.3 Root-Mean-Square Current by Numerical Integration (Electrical Engineering) 677

24.4 Numerical Integration to Compute Work (Mechanical/Aerospace Engineering) 680

Problems 684

EPILOGUE: PART SIX 694 PT6.4 Trade-Offs 694 PT6.5 Important Relationships and Formulas 695 PT6.6 Advanced Methods and Additional References 695

PART SEVEN

ORDINARY PT7.1 Motivation 699 DIFFERENTIAL PT7.2 Mathematical Background 703 EQUATIONS 699 PT7.3 Orientation 705

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CONTENTS xi

CHAPTER 25 Runge-Kutta Methods 709

25.1 Euler’s Method 710 25.2 Improvements of Euler’s Method 721 25.3 Runge-Kutta Methods 729 25.4 Systems of Equations 739 25.5 Adaptive Runge-Kutta Methods 744 Problems 752

CHAPTER 26 Stiffness and Multistep Methods 755

26.1 Stiffness 755 26.2 Multistep Methods 759 Problems 779

CHAPTER 27 Boundary-Value and Eigenvalue Problems 781

27.1 General Methods for Boundary-Value Problems 782 27.2 Eigenvalue Problems 789 27.3 Odes and Eigenvalues with Software Packages 801 Problems 808

CHAPTER 28 Case Studies: Ordinary Differential Equations 811

28.1 Using ODEs to Analyze the Transient Response of a Reactor (Chemical/Bio Engineering) 811

28.2 Predator-Prey Models and Chaos (Civil/Environmental Engineering) 818 28.3 Simulating Transient Current for an Electric Circuit (Electrical Engineering) 822 28.4 The Swinging Pendulum (Mechanical/Aerospace Engineering) 827 Problems 831

EPILOGUE: PART SEVEN 841 PT7.4 Trade-Offs 841 PT7.5 Important Relationships and Formulas 842 PT7.6 Advanced Methods and Additional References 842

PART EIGHT

PARTIAL PT8.1 Motivation 845 DIFFERENTIAL PT8.2 Orientation 848 EQUATIONS 845

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xii CONTENTS

CHAPTER 29 Finite Difference: Elliptic Equations 852

29.1 The Laplace Equation 852 29.2 Solution Technique 854 29.3 Boundary Conditions 860 29.4 The Control-Volume Approach 866 29.5 Software to Solve Elliptic Equations 869 Problems 870

CHAPTER 30 Finite Difference: Parabolic Equations 873

30.1 The Heat-Conduction Equation 873 30.2 Explicit Methods 874 30.3 A Simple Implicit Method 878 30.4 The Crank-Nicolson Method 882 30.5 Parabolic Equations in Two Spatial Dimensions 885 Problems 888

CHAPTER 31 Finite-Element Method 890

31.1 The General Approach 891 31.2 Finite-Element Application in One Dimension 895 31.3 Two-Dimensional Problems 904 31.4 Solving PDEs with Software Packages 908 Problems 912

CHAPTER 32 Case Studies: Partial Differential Equations 915

32.1 One-Dimensional Mass Balance of a Reactor (Chemical/Bio Engineering) 915

32.2 Defl ections of a Plate (Civil/Environmental Engineering) 919 32.3 Two-Dimensional Electrostatic Field Problems (Electrical

Engineering) 921 32.4 Finite-Element Solution of a Series of Springs

(Mechanical/Aerospace Engineering) 924 Problems 928

EPILOGUE: PART EIGHT 931 PT8.3 Trade-Offs 931 PT8.4 Important Relationships and Formulas 931 PT8.5 Advanced Methods and Additional References 932

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