Numerical Solution of Ordinary Differential Equations ...

1 Numerical Solution ofOrdinary Differential Equations (Part - 1)P. Sam JohnsonMay 3, 2020 Sam Johnson NIT Karnataka Mangaluru IndiaNumerical Solution of Ordinary Differential Equations (Part - 1)May 3, 20201/51 OverviewWe discuss the following important methods of solving Ordinary differentialequations of first / second s method of successive approximations(Method of successive integration)Taylor s series methodEuler s methodModified Euler s methodSam Johnson NIT Karnataka Mangaluru IndiaNumerical Solution of Ordinary Differential Equations (Part - 1)May 3, 20202/51 IntroductionA number of problems in science and technology can be formulated intodifferential Equations . The analytical methods of solving differentialequations are applicable only to a limited class of often Differential Equations appearing in physical problems do notbelong to any of these familiar types and one is obliged to resort tonumerical methods.

2 These methods are of even greater importance whenwe realize that computing machines are now readily available which reducenumerical work Johnson NIT Karnataka Mangaluru IndiaNumerical Solution of Ordinary Differential Equations (Part - 1)May 3, 20203/51 Solution of a Differential EquationThe Solution of an Ordinary Differential equation means finding an explicitexpression foryin terms of a finite number of elementary functions a Solution of a Differential equation is known as theclosedorfiniteform of the absence of such a Solution , we have Numerical methods to calculateapproximate Johnson NIT Karnataka Mangaluru IndiaNumerical Solution of Ordinary Differential Equations (Part - 1)May 3, 20204/51 Numerical Solution of Ordinary Differential Equations ofFirst OrderLet us consider the first order Differential equationdydx=f(x,y)giveny(x0) =y0(1)to study the various Numerical methods of solving such most of these methods, we replace the Differential equation by adifference equation and then solve methods yieldsolutionseither as a power series inxfrom whichthe values ofycan be found by direct substitution or a set of values Johnson NIT Karnataka Mangaluru IndiaNumerical Solution of Ordinary Differential Equations (Part - 1)May 3, 20205/51 Single-step MethodsThe methods of Picard and Taylor series belong to the former class ofsolutions.

3 In these methods,yin (1) is approximated by a truncatedseries, each term of which is a function information about the curve at one point is utilized and the Solution isnot such, these are referred to assingle-step Johnson NIT Karnataka Mangaluru IndiaNumerical Solution of Ordinary Differential Equations (Part - 1)May 3, 20206/51 Step-by-step MethodsThe methods of Euler, Runge-Kutta, Milne, Adams-Bashforth etc. belongto the latter class of these methods, the next point on the curve is evaluated in short stepsahead, by performing iterations till sufficient accuracy is achieved. Assuch, these methods are calledstep-by-step and Runge-Kutta methods are used for computingyover a limitedrange ofx-values whereas Milne and Adams methods may be applied forfindingyover a wider range Milne and Adams methods require starting values which arefound by Picard, Taylor series or Runge-Kutta Johnson NIT Karnataka Mangaluru IndiaNumerical Solution of Ordinary Differential Equations (Part - 1)May 3, 20207/51 Initial and Boundary ConditionsAn Ordinary Differential equation of thenth order is of the formF(x,y,dydx,d2ydx2.)

4 ,dnydxn)= general Solution containsnarbitrary constants and is of the form (x,y,c1,c2, ..,cn) = Johnson NIT Karnataka Mangaluru IndiaNumerical Solution of Ordinary Differential Equations (Part - 1)May 3, 20208/51 Initial / Boundary Value ProblemsTo obtain its particular Solution ,nconditions must be given so that theconstantsc1,c2,..,cncan be these conditions are prescribed at one point only (say,x0), then thedifferential equation together with the conditions constitute aninitialvalue problemof thenth the conditions are prescribed at two or more points, then the problem istermed asboundary value Johnson NIT Karnataka Mangaluru IndiaNumerical Solution of Ordinary Differential Equations (Part - 1)May 3, 20209/51 Picard s MethodCondier the first order Differential equationdydx=f(x,y)giveny(x0) =y0.

5 (2)It is required to find that particular Solution of (2) which assumes thevaluesy0whenx= (2) between limits, we get yy0dy= xx0f(x,y)dx y=y0+ xx0f(x,y) is an integral equation equivalent to (2).Sam Johnson NIT Karnataka Mangaluru IndiaNumerical Solution of Ordinary Differential Equations (Part - 1)May 3, 202010/51 Picard s MethodAs a first approximationy1to the Solution , we puty=y0inf(x,y) andintegrate (2), givingy1=y0+ xx0f(x,y0) a second approximationy2, we puty=y1inf(x,y) and integrate (2),givingy2=y0+ xx0f(x,y1) this process, we obtainy3,y4,..,yn, whereyn=y0+ xx0f(x,yn 1) this method gives a sequence of approximationsy1,y2,.., eachgiving a better result than the preceding Johnson NIT Karnataka Mangaluru IndiaNumerical Solution of Ordinary Differential Equations (Part - 1)May 3, 202011/51 Picard s MethodPicard s method is of considerable theoretical value, but can beapplied only to a limited class of Equations in which the successiveintegrations can be performed method can be extended to simultaneous Equations andequations of higher Johnson NIT Karnataka Mangaluru IndiaNumerical Solution of Ordinary Differential Equations (Part - 1)May 3, 202012/51 Exercises1.

6 Using Picard s process of successive approximations, obtain a solutionupto the fifty approximation of the equationdydx=y+xsuch thaty= 1 whenx= 0. Check your answer by finding the exactparticular Find the value ofyforx= by Picard s method, given thatdydx=y xy+xsuch thaty= 1 whenx= Johnson NIT Karnataka Mangaluru IndiaNumerical Solution of Ordinary Differential Equations (Part - 1)May 3, 202013/51 Exercises3. Solve fory( ) by Picard s method,dydx=y2+xwherey(0) = Use Picard s method to approximate the value ofywhenx= thaty(0) = 1 anddydx= 3x+ Find the Solution ofdydx= 1 +xywhich passes through (0,1) in the interval (0, ) such that the valueofyis correct to 3 decimal places. Use the whole interval as oneinterval only and takeh= Johnson NIT Karnataka Mangaluru IndiaNumerical Solution of Ordinary Differential Equations (Part - 1)May 3, 202014/51 Exercises6.

7 Use Picard s method to approximateywhenx= given thaty= 1whenx= 0 anddydx=x Find an approximate Solution of the initial value problemy = 1 +y2,y(0) = 0 by Picard s method and compare with theexact Find the successive approximate Solution of the Differential equationy =y,y(0) = 1, by Picard s method and compare it with the Given the Differential equationdydx=x2y2+ 1with the initial conditiony= 0 whenx= 0. Use Picard s method toobtainyfor , and correct to 3 decimal Johnson NIT Karnataka Mangaluru IndiaNumerical Solution of Ordinary Differential Equations (Part - 1)May 3, 202015/51 Solving Simultaneous First Order Differential Equations :Picard s MethodConsider the simultaneous Differential Equations of the typedydx=f(x,y,z)(3)anddzdx=g(x,y,z)(4)w ith initial conditionsy(x0) =y0andz(x0) =z0can be solved by Picard first approximate Solution of the Equations (3) and (4) are given byy1=y0+ xx0f(x,y0,z0)dxz1=z0+ xx0g(x,y0,z0) Johnson NIT Karnataka Mangaluru IndiaNumerical Solution of Ordinary Differential Equations (Part - 1)May 3, 202016/51 Solving Simultaneous First Order Differential Equations .

8 Picard s MethodSimilarly, the higher approximations ofyandzare given as follows:y2=y0+ xx0f(x,y1,z1)dx,z2=z0+ xx0g(x,y1,z1)dxy3=y0+ xx0f(x,y2,z2)dx,z3=z0+ xx0g(x,y2,z2)dxand so Johnson NIT Karnataka Mangaluru IndiaNumerical Solution of Ordinary Differential Equations (Part - 1)May 3, 202017/51 Exercises10. Apply Picard s method to find the second approximations to thevalues ofyandzby corresponding tox= given thatdydx=z,dzdx=x3(y+z)given thaty= 1,z=12whenx= Approximateyandzby using Picard s method for the particularsolution ofdydx=x+z,dzdx=x y2given thaty= 2,z= 1 whenx= Johnson NIT Karnataka Mangaluru IndiaNumerical Solution of Ordinary Differential Equations (Part - 1)May 3, 202018/51 Exercises12. Approximateyandzby using Picard s method for the particularsolution ofdydx= 1 +xyz,dzdx=x+y+zgiven thaty= 0,z= 1 whenx= Johnson NIT Karnataka Mangaluru IndiaNumerical Solution of Ordinary Differential Equations (Part - 1)May 3, 202019/51 Solving Second / Higher Order Differential Equations :Picard s MethodConsider a second order Differential equationd2ydx2=f(x,y,dydx).

9 Letz=dydx. Hence the given equation can be reduced to 2 first ordersimultaneous Differential equationsdydx=zanddzdx=f(x,y,z).Sam Johnson NIT Karnataka Mangaluru IndiaNumerical Solution of Ordinary Differential Equations (Part - 1)May 3, 202020/51 Exercises13. Use Picard s method to approximateywhenx= given thaty= 1,y = 0 whenx= 0 andy =y+xy .14. Using Picard s method, obtain the second approximation to thesolution atx= of the Differential equationd2ydx2=x3dydx+x3ywithy(0) = 1,y (0) = 1 Johnson NIT Karnataka Mangaluru IndiaNumerical Solution of Ordinary Differential Equations (Part - 1)May 3, 202021/51 Taylor s Series MethodConsider the first order Differential equationdydx=f(x,y)giveny(x0) =y0.(5)Differentiating (5), we haved2ydx2= f x+ f y =fx+ this successively, we can gety , , the values ofy ,y ,y.

10 ,can be obtainedatx= Johnson NIT Karnataka Mangaluru IndiaNumerical Solution of Ordinary Differential Equations (Part - 1)May 3, 202022/51 Taylor s Series MethodHence the Taylor s seriesy=y0+ (x x0)(y )x=x0+(x x0)22!(y )x=x0+ (6)gives the values ofyfor every value ofxfor which (6) finding the valuey1forx=x1from (6),y ,y etc. can be evaluatedatx= be expanded aboutx= this way, the Solution can be extended beyond the range of convergenceof series (6).Sam Johnson NIT Karnataka Mangaluru IndiaNumerical Solution of Ordinary Differential Equations (Part - 1)May 3, 202023/51 Main Drawback of Taylor s Series MethodThis is a single step method and works well so long as the successivederivatives can be calculated (x,y) is somewhat complicated and the calculation of higher orderderivatives becomes tedious, then Taylor s method cannot be usedgainfully.