A textbook of Computer Based Numerical and Statiscal Techniques part 36

A textbook of Computer Based Numerical and Statiscal Techniques part 36. By joining statistical analysis with computer-based numerical methods, this book bridges the gap between theory and practice with software-based examples, flow charts, and applications. Designed for engineering students as well as practicing engineers and scientists, the book has numerous examples with in-text solutions. | 336 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES z .2 2 z x .2 x .2 4 z x .2 5 z y .4 .21 .2 - .8536 - - - - - - 2 6 24 120 .45068 .451. Ans. PICARD S METHOD OF SUCCESSIVE APPROXIMATIONS I Consider first order differential equation dx f x y with the initial condition . 1 y yo at x x0 Integrating 1 with respect to x between x0 and x we have y x I dy 1 f x y dx yo xo or x y y01 x y dx x0 . 2 Now we solve 2 by the method of successive approximation to find out the solution of 1 . The first approximate solution approximation y1 of y is given by yi yo fx f x yo dx x0 Similarly the second approximation y2 is given by y2 yo ix f x y dx xo for the nth aproximation yn is given by yn yo fx f x yn-i dx xo . 3 with y x0 y0. Hence this method gives a sequence of approximation y1 and it can be proved f x y is bounded in some regions containing the point x0 y0 and if f x y satisfies the Lipchitz condition namely f x y -f x y k y-y where k is a constant. Then the sequence y1 to the sol. 2 . Example 6. Use Picard s method to obtain y for x . Given dy x - y with initial condition y 1 when x 0. dx NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATION 337 Sol. Here f x y x-y x0 0 y0 1 We have first approximation y1 y0 1 f x y0 dx 1 1 x -1 dx x2 2 1 - x Second approximation y2 y0 1 f x y dx 1 1 x - y fa x x2 1 1 I x 1 x--- dx 0 I 2 I 2 x3 1 - x x2---- 6 Third approximation y 3 y0 J 0x f x y 2 fa 1 J 0x x - y 2 fa 1 I x -1 x - x2 dx 06 1 - x x2 - 6 24 Fourth approximation y4 y0 J 0f x y3 dx 1 J 0 x - y3 dx 1 x3 x4 A x -1 x - x2 dx 2 1 - x x 3 4 5 x x x ------1---------- 3 12 120 Fifth Approximation y 5 y0 x .3 .4 .5 A 2 x x x x -1 x - x ---------1---idx 3 12 120 .3 .4 .5 .6 2 x x x x 1 - x x-------1--------- 3 12 60 720 When y1 .82 y4 .83746 x we get y2 .83867 y5 .83746 y3 .83740 Thus y .837 when x .2. Ans. 338 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Example 7. Find the solution of 1 xy y 0 1 which passess .