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E7_L26_Introduction_ODE_F08

# E7_L26_Introduction_ODE_F08 - 1 E7 INTRODUCTION TO COMPUTER...

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1 E7: INTRODUCTION TO COMPUTER ROGRAMMING FOR SCIENTISTS AND PROGRAMMING FOR SCIENTISTS AND ENGINEERS Lecture Outline 1. Introduction to ordinary differential equations . umerical solution of ordinary differential 2. Numerical solution of ordinary differential equations using Matlab 3. 3.Examples opyright 2007 Horowitz Packard This work is licensed under the Creative Commons Attribution- hare E7 L24 Copyright 2007, Horowitz, Packard. This work is licensed under the Creative Commons Attribution Share Alike License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/2.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA. 2 Introduction to differential equations pplying the “conservation laws” at make up a Applying the conservation laws that make up a scientific field often leads to differential equations: Mechanics: “rate of change of linear momentum of a mass particle” equals the forces applied to the particle. Thermodynamics: “rate of change of internal energy of a system” equals the net flow of energy into the system. y Chemistry: “rate of change of concentration of a E7 L24 molecule” equals a sum of terms proportional to the products of concentrations of other molecules. 3 Introduction to differential equations pplying the “conservation laws” at make up a Applying the conservation laws that make up a scientific field often leads to differential equations: Rate of change means derivative. Notation: ) y x () dy x dx first derivative of y with respect to its argument ( x ) ) dy t first derivative of with E7 L24 y y t dt = ± y respect to time t 4 Introduction to differential equations pplying the “conservation laws” at make up a Applying the conservation laws that make up a scientific field often leads to differential equations: Rate of change means derivative. Notation: ) t 2 2 d y t yt dt = ±± second derivative of y with respect to time t 3 ) third derivative of with E7 L24 3 dyt y t dt = ±±± y respect to time t

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5 Introduction to differential equations pplying the “conservation laws” at make up a Applying the conservation laws that make up a scientific field often leads to differential equations: Rate of change means derivative. Notation: [] () n n n dyx yx dx = nth derivative of y with respect to its argument ( x ) E7 L24 6 Introduction to Ordinary Differential Equations any problems in science and engineering lead to Many problems in science and engineering lead to Ordinary Differential Equations ( ODE s) of the form (, ) dy x f x y = dx where: x is the independent variable (often time) y is the dependent variable E7 L24 f(x,y) is a known function of its arguments 7 Solution of ODEs (Initial Value Problem) iven the ODE ) y x • Given the ODE dy x f xy dx = • and an initial condition 00 y = • find the function y x (slight abuse of notation) y = •w h i c h atisfies: ) y x E7 L24 satisfies: (, () ) dy x f xyx dx = 0 x x for 8 Solution of ODEs: a simple example onsider the equation that governs the decay of C 4
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E7_L26_Introduction_ODE_F08 - 1 E7 INTRODUCTION TO COMPUTER...

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