ࡱ>  Y rbjbjWW $>==n#]8LB~ (^E$iAAAAAAA$CEA}}}A-; ---}@ 8A}A--22@A* $@ A. NUMERICAL METHODS  The Rssler Attractor 5. NUMERICAL METHODS 5.0 INTRODUCTION One of the most important tasks in a study of dynamical systems is the numerical calculation of the trajectories. Thus far we have considered the integration method to be a black box into which we pop the system, initial conditions, method and time range and out pops a plot of the trajectories. Although this approach is common in courses on dynamical systems it obscures many of the pitfalls of numerical integration. It is not possible at the present state of the art to choose a best algorithm for the calculation of trajectories. There are several types of numerical algorithm, each with their own advantages and disadvantages. We shall consider a class of methods known as discrete variable methods. These methods approximate the continuous time dynamical system by a discrete time system. This means that we are not really simulating the continuous system but a discrete system which may have different dynamical properties. This is an extremely important point. The discrete variable methods which we consider fall into two main types, Runge-Kutta methods and Linear Multistep methods. Maple has implementations of both types of method as well as a number of more sophisticated techniques designed to overcome some of the pitfalls of numerical solution. The more sophisticated methods still fall into the discrete variable category. 5.1 TYPES OF METHOD Although the dynamical systems which we are simulating are usually in more than one dimension we can, without loss, restrict our numercal anlaysis of the methods to the single non-autonomous differential equation  EMBED Equation  subject to the initial condition  EMBED Equation . We shall usually refer to the differential equation together with the initial condition as an initial value problem (IVP). Discrete variable methods yield a series of approximations  EMBED Equation  on the set of points  EMBED Equation  where h is the stepsize. Taylor Series Method These methods are based on the Taylor series expansion  EMBED Equation  If the series is truncated and  EMBED Equation  is replaced by the approximation  EMBED Equation  we obtain the Taylor Series Method of order p  EMBED Equation  where  EMBED Equation . Although there is an implementation of this method in Maple it is not much used in practice due to the necessity of computing the higher order derivatives of X. We shall only use it as a reference method when discussing the accuracy of other methods. Runge-Kutta Methods These methods are based on the notion of finding a formula which agrees with the Taylor series as closely as possible without involving derivatives. For example consider the possibility of matching the second order Taylor series method  EMBED Equation  by using a formula of the form  EMBED Equation  where  EMBED Equation  The equivalent Taylor series expression to  EMBED Equation  is  EMBED Equation  Expanding  EMBED Equation  in a Taylor series up to terms of  EMBED Equation  gives  EMBED Equation  Comparing the two expressions we see that  EMBED Equation  resulting in the family of solutions  EMBED Equation  where  EMBED Equation  is a free parameter. For  EMBED Equation  we obtain the improved Euler method  EMBED Equation  and for  EMBED Equation  the modified Euler method  EMBED Equation  Unfortunately the terminology for naming second order Runge-Kutta methods is not standardised and Maple calls the improved Euler method the Heun formula and the modified Euler method the improved polygon method. The procedure above can be extended to give higher order methods such as the classical 4th order method  EMBED Equation  Linear Multistep Methods These methods are based on integration of an interpolating polynomial having the values  EMBED Equation  on a set of points  EMBED Equation  at which  EMBED Equation  has already been computed. By integrating  EMBED Equation  over the interval  EMBED Equation  we obtain  EMBED Equation  Using linear interpolation in the interval we have  EMBED Equation  Integrating gives  EMBED Equation  This leads to the Trapezoidal method  EMBED Equation  The general form of these methods is  EMBED Equation  where  EMBED Equation  and  EMBED Equation  are constants,  EMBED Equation . This formula is called a linear k-step method. In order to generate the sequence of approximations  EMBED Equation  it is first necessary to obtain k starting values  EMBED Equation . If  EMBED Equation  the method is explicit. If  EMBED Equation  then the method is implicit and leads to a non-linear equation for  EMBED Equation . The main methods of this type which we shall consider are: Adams-Bashforth These methods are explicit with methods of order k being k-step. Methods of order from one to three have the formulae  EMBED Equation  The first order method is more normally called the Euler method. Adams-Moulton These methods are implicit with methods of order k being  EMBED Equation -step. Methods of order from one to three have the formulae  EMBED Equation  The first order method is called the backward Euler formula and the second order method is the Trapezoidal method. Gear Methods These methods are implicit with methods of order k being k-step. Methods of order from one to three have the formulae  EMBED Equation  The first order method is again the backward Euler formula. 5.2 MAPLE IMPLEMENTATION Maple contains a large range of numerical procedures for integrating non-autonomous dynamical systems of the type  EMBED Equation  All of these procedures are invoked using the Maple dsolve command with the numeric option. The syntax for the command is dsolve(deqns, vars, type=numeric, options) where deqns defines the system of differential equations and initial values, vars defines the dependent and independent variables, type=numeric tells Maple to use a numerical algorithm and options allows a choice of method, stepsize and other options associated with the method. The default method is a Fehlberg fourth-fifth order Runge-Kutta method. Worked Example 1 - Van der Pol Equation As an example consider the solution of the Van der Pol equation written as the first order system  EMBED Equation  > epsilon:=10: > ivp := {diff(x(t),t)=y(t),diff(y(t),t)=-x(t)+epsilon*(1-x(t)^2)*y(t), x(0)=0,y(0)=0.5}: > fcns := {x(t), y(t)}: This defines the initial value problem. Now we invoke dsolve > p1:= dsolve(ivp,fcns,type=numeric);  EMBED Word.Picture.8  Note that the output from the dsolve command is a procedure. In order to find the numerical solution we need to evaluate the procedure at the appropriate value of t. > p1(10);  EMBED Word.Picture.8  We can us the Maple odeplot command to graph the solution. This command is in the plots package so this has to be loaded first. > with(plots): Now we can plot the solution in various ways. Firstly here is a plot of the solution components v t over the range [0,20]. numpoints controls the number of plotted points which needs to be relatively high here to obtain a realistic plot. > odeplot(p1,[[t,x(t)],[t,y(t)]],0..20, numpoints=500);  It is also possible to obtain a phase plot > odeplot(p1,[[x(t),y(t)]],0..20, numpoints=500);  and a three dimensional plot > odeplot(p1,[[t,x(t),y(t)]],0..20, numpoints=500);  Classical Methods Maple contains a number of one-step methods for the numerical solution of initial value problems. These are referred to as classical methods and are invoked by including the option method=classical[type] in the call to dsolve. Here type can be one of foreuler the forward Euler method; heunform the Heun formula (also known as the trapezoidal rule, or the improved Euler method); impoly the improved polygon method (also known as the modified Euler method); rk2 the second-order classical Runge-Kutta method; rk3 the third-order classical Runge-Kutta method; rk4 the fourth-order classical Runge-Kutta method; adambash the Adams-Bashford method (a "predictor" method); abmoulton the Adams-Bashford-Moulton method (a "predictor- corrector" method). If no type is specified the forward Euler method is used. Worked Example 2 - The Forward Euler Method Consider the IVP  EMBED Equation  We can define this in Maple as > ivp:={diff(y(x),x)=-2*x*y(x)^2,y(0)=1}: In this case Maple can find the exact solution using dsolve > Exactsoln:=rhs(dsolve(ivp,y(x)));  EMBED Word.Picture.8  Now we use Euler's method to obtain the numerical solution. Note that this method like all the other methods of type classical uses a fixed stepsize which we provide. > es0:=dsolve(ivp,y(x),type=numeric, method=classical[foreuler],stepsize=0.001): Thus we can find the solution at  EMBED Equation  > es0(0.4);  EMBED Word.Picture.8  and plot the solution for a range of values of x > odeplot(es0,[x,y(x)],0..6,labels=[x,y]);  Now often we want to investigate how the solution behaves for differing values of the stepsize h. In order to do this we can define a different version of the function for the numerical solution which uses the stepsize as one of the input parameters > es1:=h->dsolve(ivp,y(x),type=numeric, method=classical[foreuler],stepsize=h): Thus we can obtain the solution for different stepsizes > es1(0.001)(0.4);  EMBED Word.Picture.8  > es1(0.01)(0.4);  EMBED Word.Picture.8  We can compare the solution at different stepsizes by constructing a table of values as follows: Firstly define the output points > x:=k->k*0.1: Now define a function which finds the approximate solution at a given output point > EulerSoln:=(x,h)->rhs(es1(h)(x)[2]): Define the exact solution > ExactSoln:=x->1/(1+x^2);  EMBED Word.Picture.8  Construct an array whose elements compare the exact solution to the numerical solution for three different stepsizes. > mm:=array(1..8,1..5): mm[1,1]:=`x(k)`:mm[1,2]:=`Exactsoln`:mm[1,3]:=`h=0.1`: mm[1,4]:=`h=0.01`:mm[1,5]:=`h=0.001`: for i from 2 to 8 do mm[i,1]:=0.1*(i-2): mm[i,2]:=evalf(ExactSoln(x(i-2)),5): for j from 3 to 5 do mm[i,j]:=evalf(EulerSoln(x(i-2),10^(-j+2)),5) od: od: > eval(mm);  EMBED Word.Picture.8  Another possibility is to compare the errors at each stepsize. Firstly define a function giving the error > err:=(x,h)->ExactSoln(x)-EulerSoln(x,h): > tt:=array(1..8,1..4): tt[1,1]:=`x(k)`:tt[1,2]:=`h=0.1`:tt[1,3]:=`h=0.01`:tt[1,4]:=`h=0.001`: for i from 2 to 8 do tt[i,1]:=0.1*(i-2); for j from 2 to 4 do tt[i,j]:=evalf(err(x(i-2),10^(-j+1)),5); od: od: > eval(tt);  EMBED Word.Picture.8  Worked Example 3 - The Classical Second Order Runge-Kutta Method Consider the solution of the IVP  EMBED Equation  > IVP:={diff(y(x),x)=-4*y(x)+4*x,y(0)=1}: The exact solution is given by > dsolve(IVP,y(x));  EMBED Word.Picture.8  Use the 2nd order classical Runge-Kutta method > rk2:=h->dsolve(IVP,y(x),type=numeric,method=classical[rk2],stepsize=h): > x:=k->k*0.5: > RK2Soln:=(x,h)->rhs(rk2(h)(x)[2]): > ExactSoln:=x->x-1/4+5/4*exp(-4*x);  EMBED Word.Picture.8  > mm:=array(1..10,1..5): mm[1,1]:=`x(k)`:mm[1,2]:=`Exactsoln`:mm[1,3]:=`h=0.25`: mm[1,4]:=`h=0.5`:mm[1,5]:=`h=0.75`: for i from 2 to 10 do mm[i,1]:=0.5*(i-2): mm[i,2]:=evalf(ExactSoln(x(i-2)),5): for j from 3 to 5 do mm[i,j]:=evalf(RK2Soln(x(i-2),0.25*(j-2)),5) od; od: > eval(nm);  EMBED Word.Picture.8  Note that as the stepsize is increased the numerical solution fails to represent the exact solution accurately. Indeed for a stepsize of 0.75 the numerical solution 'blows up'. This is due to non-convergence as a result of the numerical method becoming unstable. We shall consider this phenomenon next. Exercises 1 1. Use the classical numerical methods foreuler, heunform, rk3, rk4 and adambash to attempt to obtain a numerical solution of the IVPs (a)  EMBED Equation  (b)  EMBED Equation  Use a range of stepsizes in the interval [0,1]. At what approximate value of the stepsize do the methods become unstable. 2. Use each of the methods above to solve the systems of differential equations (a)  EMBED Equation  (b)  EMBED Equation  where  EMBED Equation  is a parameter. (Try  EMBED Equation ). (c)  EMBED Equation  In each case use odeplot to obtain time series and phase plots. 5.3 LOCAL AND GLOBAL ERRORS The output of a discrete variable method is a set of points  EMBED Equation and the output of the dynamical system is a continuous trajectory  EMBED Equation . For the numerical results to provide a good approximation to the trajectory we require that the difference  EMBED Equation  where  EMBED Equation  is some defined error tolerance, at each solution point. This difference is called the global error and is the accumulated error over all solution steps. Unfortunately it is extremely difficult to accomplish this and we have to confine ourselves to controlling the local error  EMBED Equation  at each step where  EMBED Equation  is the numerical solution obtained on the assumption that the numerical solution at the previous solution point is exact. There are two sources of local error,the roundoff error and the truncation error. Roundoff Error The roundoff error is the error which arises from the fact that numerical methods are implemented on digital computers which only calculate results to a fixed precision which is dependent on the computer system used. Note that since roundoff errors depend only on the number and type of arithmetic operations per step and is thus independent of the integration stepsize h. Truncation Error The truncation error of a numerical method results from the approximation of a continuous dynamical system by a discrete one. The truncation error is machine independent, depending only on the algorithm used and the stepsize h. An important concept in the analysis of the truncation error is that of consistency. Basically consistency requires that the discrete variable method becomes an exact representation of the dynamical system as the stepsize  EMBED Equation . Consistency conditions can be derived for both Linear Multistep and Runge-Kutta methods. Linear Multistep Methods Consider the general linear multistep method  EMBED Equation  EMBED Equation  We can define the first characteristic poynomial by  EMBED Equation  and the second characteristic polynomial by  EMBED Equation  We can show that consistency requires that  EMBED Equation  Runge-Kutta Methods The general pth order Runge-Kutta method can be written in the form  EMBED Equation  Here we have  EMBED Equation  and it can be shown that consistency requires that  EMBED Equation  Worked Example 4 Examine the consistency of (a) the classical 4th order Runge-Kutta method, (b) the two-step Adams-Bashforth method.  EMBED Equation  thus  EMBED Equation  and hence the method is consistent.  EMBED Equation  thus  EMBED Equation  and hence the method is consistent. The accuracy with which a consistent numerical method represents a dynamical system is determined by the order of consistency. The method of determining this is best illustrated by an example. Worked Example 5 Determine the order of consistency of the Trapezoidal method. The order of consistency is determined by substituting the exact solution  EMBED Equation  into the formula of the numerical algorithm and expanding the difference between the two sides of the formual by Taylor series. The result is then normalised by multiplying by the scaling factor  EMBED Equation .  EMBED Equation  thus  EMBED Equation  and the method is consistent. Now the truncation error is given by  EMBED Equation  The order is given by the highest power of h remaining. Hence the method is consistent of order two. 5.4 ZERO STABILITY This is a problem peculiar to consistent linear k-step methods in which a first order dynamical system is integrated using a kth order difference equation. This leads to the possible existence of spurious solutions of the difference equation which can swamp the desired solution. In order to avoid this occuring we have to restrict the roots of the first characteristic polynomial  EMBED Equation  to satisfy the root condition. Definition Root Condition We say that a linear k-step method satisfies the root condition if the roots of the characteristic polynomial  EMBED Equation  all lie within or on the unit circle, those on the unit circle being simple. Note that the roots of  EMBED Equation  may be complex hence the necessity of considering the unit circle rather than the interval  EMBED Equation  in the definition. Theorem Zero-Stability A a linear k-step method is zero stable if and only if it satisfies the root condition. We can now state the fundamental theorem concerning convergence: Theorem - Convergence A discrete variable method is convergent if and only if it is both consistent and zero stable. Often it is desirable for the roots of  EMBED Equation  to satisfy the strong root condition. Definition Strong Root Condition A linear k-step method is said to satisfy the strong root condition if the characteristic polynomial has a simple root at  EMBED Equation  and all the remaining roots lie strictly within the unit circle. The roots  EMBED Equation  of  EMBED Equation  for a consistent method satisfying the root condition can be categorized as  EMBED Equation  Worked Example 6 Show that the Gear method  EMBED Equation  is convegent. Before determining the characteristic polynomials write in the standard form  EMBED Equation  Then  EMBED Equation  Checking consistency  EMBED Equation  The roots of  EMBED Equation  are given by  EMBED Equation  and hence the method is zero-stable. Combining these results we can conclude that the method is convergent. Exercises 2 1. (a) Show that the method  EMBED Equation.3  is consistent and determine the value of  EMBED Equation.3  which maximizes the order of the method. Find the range of values of  EMBED Equation.3  for which the method is zero stable. 2. Show that Quades method  EMBED Equation.3  is both consistent and zero stable. Show that the 3-step Gear method  EMBED Equation.3  is both consistent and zero stable. ABSOLUTE STABILITY So far we have considered the behaviour of numerical methods in the limit as the stepsize  EMBED Equation . However in practice we must deal with finite stepsizes. To illustrate the problems that might arise consider the mid-point method  EMBED Equation  This is a linear two-step method. In standard form the method is  EMBED Equation  thus  EMBED Equation  Checking consistency  EMBED Equation  The roots of  EMBED Equation  are given by  EMBED Equation  hence the method is both consistent and zero-stable and hence convergent. Now consider the solution of the initial value problem  EMBED Equation  by the mid-point method using a stepsize  EMBED Equation . Using Maple we define the initial value problem > f:=(t,x)->-2*t*x(t)^2:x0:=1: > IVP:={diff(x(t),t)=f(t,x),x(0)=x0}: > FCN:={x(t)}: We can find the exact solution > Exact:=rhs(dsolve(IVP,FCN));  The mid-point method requires a starting value which can be obtained from the classical fourth order Runge-Kutta method > sv:=h->dsolve(IVP,FCN,type=numeric,method=classical[rk4],stepsize=h): > SV:=(t,h)->rhs(sv(h)(t)[2]): Now define the mid-point method > midpt:=proc(n,h) option remember; if n=1 then SV(h,h) elif n=2 then x0+2*h*f(h,SV(h,h)) else midpt(n-2,h)+2*h*f((n-1)*h,midpt(n-1,h))fi; end: Obtain a plot of the solution > plotmidpt:=proc(N,h) local l,i; l:=[]; for i from 1 to N do l:=[op(l),i*h,midpt(i,h)]; od; pointplot(l,connect=true); end: Finally plot both the numerical and exact solutions > with(plots): > p1:=plotmidpt(50,0.1): > p2:=plot(Exact,t=0..5): > display({p1,p2});  Notice that the numerical solution becomes increasingly innacurate, oscillating about the exact solution, as t increases. This behaviour arises because the behaviour of the numerical solution does not mimic that of the exact solution. In this case the problem arises because of a spurious solution of the difference equation corresponding to the root  EMBED Equation  of  EMBED Equation . However the problem can also arise in one-step methods which have no spurious solutions. Consider the Linear Multistep method.  EMBED Equation  applied to the test equation  EMBED Equation  On substitution into the method we obtain  EMBED Equation  Thus  EMBED Equation  Let  EMBED Equation , then  EMBED Equation  Hence  EMBED Equation   EMBED Equation  is called the stability polynomial of the method. Now one of the roots  EMBED Equation  will correspond to the true solution, the other roots will lead to spurious solutions whose magnitude will have to be controlled to obtain stability. Definition Absolute Stability A numerical method is said to be absolutely stable for a given  EMBED Equation if all the roots of  EMBED Equation  lie within the unit circle. A region  EMBED Equation  of the complex plane is said to be a region of absolute stability if the method is stable for all  EMBED Equation in  EMBED Equation . Worked Example 7 Find and sketch the region of absolute stability for (a) Euler's method, (b) Trapezoidal method. (a) For Euler's method  EMBED Equation  Thus  EMBED Equation   EMBED Equation  is shown below  EMBED Word.Picture.8  (b) For the Trapezoidal method  EMBED Equation  Thus  EMBED Equation  giving the region EMBED Equation  shown below  EMBED Word.Picture.8  For Runge-Kutta methods the stability polynomial has the form  EMBED Equation  where  EMBED Equation  is a polynomial for an explicit method and a rational function for an implicit method. Worked Example 8 Find and sketch the absolute stability region for the second order Runge-Kutta method  EMBED Equation  where  EMBED Equation  Substituting into the test equation  EMBED Equation  Thus  EMBED Equation  Hence the stability polynomial is given by  EMBED Equation  For absolute stability we require that  EMBED Equation  In order to draw the region of absolute stability consider the boundary EMBED Equation  of  EMBED Equation . The locus of this boundary will be the set of complex numbers z such that  EMBED Equation  Thus  EMBED Equation  In order to obtain the region we need to plot the roots of the quadratic equation  EMBED Equation  for  EMBED Equation  in the range  EMBED Equation . This is best done on a computer. The resulting stability region is shown below:  The method outlined above is an example of the boundary locus method which is easily implemented for Linear Multistep methods as follows. The stability polynomial is  EMBED Equation  and hence  EMBED Equation  but on EMBED Equation   EMBED Equation  Hence the locus of the boundary  EMBED Equation  is given by the set of complex numbers z satisfying  EMBED Equation  Worked Example 9 Find the region of absolute stability for the Gear method  EMBED Equation  From Worked Example 6  EMBED Equation  Thus the stability polynomial is given by  EMBED Equation  Substituting  EMBED Equation  and solving  EMBED Equation  Now substitute  EMBED Equation  to obtain  EMBED Equation  which gives the plot  In order to determine whether  EMBED Equation  is the interior or exterior of the closed curve choose a point inside the curve, say  EMBED Equation  and evaluate the roots of  EMBED Equation .  EMBED Equation  and we see that on of the roots,  EMBED Equation , has modulus greater than one and hence  EMBED Equation  must consist of the exterior of the closed curve. Worked Example 10 Show that the mid-point method  EMBED Equation  has an empty region of absolute stability. From above  EMBED Equation  Thus the stability polynomial is given by  EMBED Equation  Substituting  EMBED Equation  and solving  EMBED Equation  Now substitute  EMBED Equation  to obtain  EMBED Equation  which does not bound any region of the complex plane. Hence  EMBED Equation  is empty. Exercises 3 1. Show, using the boundary locus method, that Quades method  EMBED Equation.3  has no real region of absolute stability. 2. Determine the regions of absolute stability of the methods (a)  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  APPENDIX 1 ANSWERS AND HINTS FOR THE EXERCISES Methods Exercises 1 Use the method outlined in section to obtain tables comparing the exact and numerical solutions for a variety of stepsizes. Use the method outlined in section to obtain solutions and plots with the odeplot command. Consistency and Zero Stability Exercises 2 (a) Truncation error is  EMBED Equation  hence maximum order when  EMBED Equation . Method is zero stable if  EMBED Equation .  EMBED Equation , hence  EMBED Equation  and roots of  EMBED Equation  are  EMBED Equation  which all have absolute value one.  EMBED Equation , hence  EMBED Equation  and roots of  EMBED Equation  are  EMBED Equation  which all have absolute value less than or equal to one. Absolute Stability Exercises 3 Boundary locus gives  EMBED Equation  (a) Region inside  (b) Region inside  Region outside  Region outside  APPENDIX 2 MAPLE WORK SHEETS Worked Example 5 - Determining The Order of a Numerical Method > restart: Define a function to represent the exact solution > x:=t->x(t): Use some aliases for the derivatives of x > alias(x1=D(x),x2=(D@@2)(x),x3=(D@@3)(x),x4=(D@@4)(x), x5=(D@@5)(x),x6=(D@@6)(x),x7=(D@@7)(x),x8=(D@@8)(x)): Use the taylor command to obtain a Taylor series expansion of x(tn+h) > ts:=taylor(x(t),t=tn,9): Replace t-tn by h in the expansion > tsh:=subs(t-tn=h,ts): Convert to a polynomial so that we can perform the algebra later > p:=unapply(convert(tsh,polynom),h): Repeat above for x'(tn+h) > dts:=taylor(x1(t),t=tn,8): > dtsh:=subs(t-tn=h,dts): > dp:=unapply(convert(dtsh,polynom),h): Euler's Method > simplify(p(h)-x(tn)-h*x1(tn)); Trapezoidal Rule > simplify(p(h)-x(tn)-h*(x1(tn)+dp(h))/2); Quade's Method > simplify(p(4*h)-8*(p(3*h)-p(h))/19-x(tn)-6*h*(dp(4*h)+4*dp(3*h)+4*dp(h)+x1(tn))/19); Worked Example 9 > restart: > with(plots): Define the characteristic polynomials > rho:=theta->theta^2-4*theta/3+1/3; > sigma:=theta->2/3*theta^2; and the stability polynomial > pi:=theta->rho(theta)-lambda*h*sigma(theta); Substitute z = h*lambda; > piz:=theta->rho(theta)-z*sigma(theta); Define the boundary of the region > rat:=solve(piz(theta),z); > rat1:=subs(theta=exp(I*phi),rat); Plot the boundary > complexplot(rat1,phi=0..2*Pi,numpoints=500); Check whether region is inside or outside closed curve > rs:=solve(subs(z=1,piz(theta))=0,theta); Must be outside. Worked Example 10 > restart: > with(plots): Define the characteristic polynomials > rho:=theta->theta^2-1; > sigma:=theta->2*theta; Find the stability polynomial > pi:=theta->rho(theta)-lambda*h*sigma(theta); > piz:=theta->rho(theta)-z*sigma(theta); Solve for z > rat:=solve(piz(theta),z); Substitute theta = e^(i*phi); > rat1:=subs(theta=exp(I*phi),rat); Simplify the result > simplify(rat1); Thus empty region of absolute stability. 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Ole10ItemName%_987068061($ F  Ole & x(t n+1 )=x(t n )+h2x(t n )+h 2 2!2x(t n )+h 3 3!2x(t n )+Kࡱ> Allࡱ> ࡱ>  F Equation Equation EquatiCompObjive   #&#'S ObjInfomName)*+,./06789)@Ole10NativeIKJLMNPOQ%' F*DbOle10ItemNameikmlnoprwvxyz,on9qࡱ> ࡱ> @ x(t n )- 2 ࡱ> Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> _987068064     * Fe Ole$%&'()*+,-./0 6798;-BCompObjHGIJKNLMPOQ),}WXZY.SaObjInfoghikjlmnpoqvxwy{0t9ࡱ>   X nࡱ>Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ>  X n+1 =X n +h2XOle10Native+-1$Ole10ItemName2_987068066eF0 FdOle0ItemName L*3CompObj9/2 F4SObjInfo6Ole10Native137Ole10ItemName< n +h 2 2!2X n +h 3 3!2X n +K+h p p!X n(p)ࡱ> Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> _987068069e   6 F`'S OlenfomName)*+,./0 6789=@CompObjiveIKJLMNPOQ58 F>SbObjInfomNameikmlnoprwvxyz@ࡱ> ` X (p) d p Xdt pࡱ>Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ> Ole10Native79 FAdOle10ItemNameC_9870680714@< F`Q`QSOlenfo DCompObjive;>ESObjInfomNameGOle10Native=? FHOle10ItemNameK X n+1 =X n +h2X n +h 2 2!2X nࡱ> Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ> _987068074e   B F@0`7D Ole0ItemName)*+,./0 6789L@CompObj1HIKJLMNPOQAD FMSbObjInfoghjikmlnoprwvxyzOOle10NativeCEPOle10ItemNameS_987068076e:RH F@HQHOle0ItemName T X n+1 =X n +hf(t n ,X n ,h)ࡱ> Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ> CompObj4     GJ FUS ObjInfo&'()*+,-./06798;WBOle10NativeIJKNLMPOQIK}WXZYXaOle10ItemNamekjlmnpoqvxwy{[t9 f(t,x,h)=af(t,x)+bf(t+ah,x+bhf(t,x))ࡱ> Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ> _987068079e   N Fj=s'S OlenfomName)*+,./0 6789\@CompObjiveIKJLMNPOQMP F]SbObjInfomNameikmlnoprwvxyz_Ole10NativeOQ`DOle10ItemNameb_987068081LXT FOle c@ f(t,x,h) 2 `= 2 ࡱ> Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ>  D(t,x,h)=2x(t)+CompObjive   SV#dS ObjInfomName)*+,./06789f@Ole10NativeIKJLMNPOQUW FgbOle10ItemNameikmlnoprwvxyzlh22x(t)  =f(t,x)+h2f t (t,x)+f x (t,x)f(t,x)[]ࡱ> Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> _987068084 Z Fp`Ole$(+/ 589mCompObjFIJNY\VYnSObjInfohijkpࡱ> @ f(t,x,h)ࡱ> Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ> Ole10Native   []#qD Ole10ItemName)*+,./06789s@_987068086HIKJLMNPOQY` F9@a^`bOledfghjikmlnopr wvxyztCompObj9e   _b FuS ObjInfomName)*+,./06789w@Ole10NativeIKJLMNPOQac Fx$bOle10ItemNameikmlnoprwvxyzy  Oh()ࡱ>Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ>  f(t,x,h)=(a+b)f(t,x)+hbaf t (t,x)+bf_987068089e   f F@D Ole0ItemName)*+,./0 6789z@CompObj1HIKJLMNPOQeh F{SbObjInfoghjikmlnoprwvxyz}Ole10Nativegi~Ole10ItemName_987068091edpl F]dOle0ItemName L* x (t,x)f(t,x)[]+O(h 2 )h2ࡱ>Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ> ` a+b=1, ba= 12 , CompObjiveknSObjInfomNameOle10Nativemo FdOle10ItemNamebb= 12` 2 ࡱ>Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ> ` a=1-l, b=l,  a=b=12l?ࡱ>_987068094e   r F'S OlenfomName)*+,./0 6789@CompObjiveIKJLMNPOQqt FSbObjInfomNameikmlnoprwvxyzOle10NativesudOle10ItemName_987068096jx F`+`+Ole Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ>   l0ࡱ>Allࡱ> ࡱ> CompObjivewzSObjInfomNameOle10Nativey{ F$Ole10ItemNameL*_987068099 ~ F@<`IDOle$(+/ 589CompObjFIJN}VYSObjInfohijk F Equation Equation Equation9qࡱ> ࡱ>   l= 12ࡱ>Allࡱ> ࡱ>  FMathType Equation Equation Equation9qࡱ> Ole10Native$Ole10ItemName_987162273 F@U:^Ole CompObjive\ObjInfomNameOle10Native F(Ole10ItemNameࡱ> $ X n+1 =X n +h2f(t n ,X n )+ft n +h,X n +hf(t n ,X n )()[]lDlDࡱAllࡱ> ࡱ> _987068104e   | Fov'S OlenfomName)*+,./0 6789@CompObjiveIKJLMNPOQ FSbObjInfomNameikmlnoprwvxyz F Equation Equation Equation9qࡱ> ࡱ>   l=1np 2 ࡱ>Allࡱ> ࡱ>  FMathType Equation Equation Equation9qࡱ> Ole10Native F$Ole10ItemName_987162298 FlSOlenfo CompObjive   #\ ObjInfomName)*+,./06789@Ole10NativeIKJLMNPOQ FbOle10ItemNameikmlnoprwvxyzࡱ>  X n+1 =X n +hft n + h2 ,X n + h2 f(t n ,X n )()ࡱ>Allࡱ> ࡱ>  F Equation Equation Equati_987068109  F]Ole$(+/ 589CompObjFIJNVYSObjInfohijkon9qࡱ> ࡱ> @ X n+1 =X n +h6(k 1 +2k 2 +2k 3 +k 4 )k 1 =f(t n ,X n )k 2 =f(t n + 12 h,X n + 12 hk 1 )k 3 =f(tOle10Native   #D Ole10ItemName)*+,./06789@_987068111HIKJLMNPOQv F@`^`bOledfghjikmlnopr wvxyz n + 12 h,X n + 12 hk 2 )k 4 =f(t n +h,X n +hk 3 ) ࡱ> Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> CompObjiveSObjInfomNameOle10Native FdOle10ItemNameࡱ> ` f n f(t n ,X n )2 6np 2 ࡱ>Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ> _987068114e F@Y$Ole0ItemName CompObj8 FSObjInfoOle10Native$Ole10ItemName_987068116e FIHOle0ItemName   t nࡱ>Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ>   X n+%2ࡱ>Allࡱ> CompObjiveSObjInfomNameOle10Native F$Ole10ItemName_987068119  FU|Ole$(+/ 589CompObjFIJNVYSObjInfohijkࡱ>  F Equation Equation Equation9qࡱ> ࡱ> @ 2x=ft,x()esa%2P&Nࡱ> Allࡱ> ࡱ> Ole10Native   #D Ole10ItemName)*+,./06789@_987068121HIKJLMNPOQ FE/6^`bOledfghjikmlnopr wvxyzCompObjiveSObjInfomNameOle10Native FDOle10ItemName F Equation Equation Equation9qࡱ> ࡱ> @ t n ,t n+1 []ࡱ> Allࡱ> ࡱ>  F Equation Equation Equati_987068124 FG`POle CompObjSObjInfoon9qࡱ> ࡱ>  xt n+1 ()-xt n ()=f(t,x)dt t n t n+1 ࡱ> Allࡱ> ࡱ> Ole10NativeOle10ItemName_987068127e F@a`AidOle0ItemName L*CompObjive FSObjInfomNameOle10Native FdOle10ItemName F Equation Equation Equation9qࡱ> ࡱ> ` xt n+1 ()-xt n ()@t-t n+1 t n -t n+1 f n +t-t n t n+1 -t n f n+1 []dt t n t n+1 ࡱ>Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ>  xt n+1 ()-xt n ()@h2f n +_987068129 F2 ӊOle CompObjSObjInfo)      "!$#&%('*km,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijlnoqprustwvxy{z}|~Ole10Native    F Ole10ItemName)*+,./06789@_987068131eIKJLMNPOQ^ FädSbOlenfomNameikmlnopr wvxyz !'*.15;AGJKLMNOPQRSW]`abcdefghimpqrstuvwx|f n+1 ()ࡱ>Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ>  X n+1 =X n +h2f n +f n+1 ()2 CompObjiveSObjInfomNameOle10Native FOle10ItemName y ࡱ> Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ>  a j X n+jj=0k  =hb j f n+j_987068134 FUOle  CompObj SObjInfo Ole10NativeOle10ItemName_987068136 F@Ole j=0k  , n=0,1,2,K??ࡱ> Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ>   a j ࡱ>CompObj4 FSObjInfoOle10Native$Ole10ItemNameAllࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ>   b j ࡱ>Allࡱ> ࡱ> _987068139  F @QOle CompObjSObjInfoOle10Native$Ole10ItemName_987068141e F AHOle0ItemName CompObjive SObjInfomName"Ole10Native F#$Ole10ItemName$ F Equation Equation Equation9qࡱ> ࡱ>   a k 0fࡱ>Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> _987068144T 55 F +2@IOle@wG 0  wGpH%CompObjent``6&SObjInfol 0 (,@ࡱ> @ X n {}ࡱ> Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ> Ole10Native)DOle10ItemName+_987068147e FtCLdOle0ItemName L*,CompObj4 F-SObjInfo/Ole10Native0dOle10ItemName2` X 0 ,X 1 ,K,X k-1ࡱ>Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ>   b k =0ࡱ>_987068149T 55 F`e]eLOle@wG 0  wGpH3CompObjent``64SObjInfol 0 6,@Ole10Native   #7$ Ole10ItemName)*+,./067898@_987068152HIKJLMNPOQ$ F@@^`bOledfghjikmlnopr wvxyz9Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ>   b k 0ࡱ>Allࡱ> ࡱ> CompObjive:SObjInfomName<Ole10Native F=$Ole10ItemName>_987068154T 55 F*OOle@wG 0  wGpH?CompObjent``6@SObjInfol 0 B,@ F Equation Equation Equation9qࡱ> ࡱ>   X n+kࡱ>Allࡱ> ࡱ>  FMathType Equation Equation Equation9qࡱ> Ole10NativeC$Ole10ItemNameD_987075907 FOle ECompObjive  FF\ObjInfomNameHOle10Native   FIOle10ItemNameTࡱ>  X n+1 =X n +hft n ,X n ()X n+1 =X n +h23ft n ,X n ()-ft n-1 ,X n-1 (){}X n+1 =X n +h1223ft n ,X n ()-16ft n-1 ,X n-1 ()+5ft n-2 ,X n-2 (){}ࡱ> Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> _987068159 F`Ole UCompObj VSObjInfoXࡱ>   k-1()ࡱ>Allࡱ> ࡱ>  FMathType Equation Equation Equation9qࡱ> ࡱ>  X n+1 =X n +hfOle10NativeY$Ole10ItemNameZ_987076007 F` >Ole [CompObjive\\ObjInfomName^Ole10Native F_Ole10ItemNamejt n+1 ,X n+1 ()X n+1 =X n +h2ft n+1 ,X n+1 ()+ft n ,X n (){}X n+1 =X n +h125ft n+1 ,X n+1 ()+8ft n ,X n ()-ft n-1 ,X n-1 (){}ࡱ> Allࡱ> ࡱ>  FMathType Equation Equation Equation9qࡱ> ࡱ> @ X n+1 =X n +hf_987076084e F \\OlenfomName kCompObjive Fl\ObjInfomNamenOle10NativeoDOle10ItemNamey_987068167e  F79?dOle0ItemName L*zt n+1 ,X n+1 ()X n+1 = 13 4X n -X n-1 {}+2h3ft n+1 ,X n+1 ()X n+1 = 111 18X n -9X n-1 +2X n-2 {}+6h11ft n+1 ,X n+1 ()ࡱ> Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ>  "x=ft,x(), xt 0 ()=x 0CompObjive   "#{S ObjInfomName)*+,./06789}@Ole10NativeIKJLMNPOQ!# F~bOle10ItemNameikmlnoprwvxyzࡱ> Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ> ` 2x=y2y=-x+e1-x 2 ()+yࡱ>2lv t .l2L~`!lv t .l2L~@v |wxRMKQ=NAMAKk!^pSmEES J;IhJD!P}7dJt wy7`02Pcd" !ζhE#L/c$xd.U\ 蘗 NgIjYvd~pKg*wQ+  UF˳\Ŷ* A˭5{_t?V"-Zj.CUOYy^t̲D[9p(sjuNw< UZ;bS!W ߯Z-@aDtM:oN pC+j\F}YļVu<u߼wPPDd|<  C A26|4Jrƈrp^8$9`!6|4Jrƈrp^8$`L 0bxuQMKQ=c9JNBRZDZZBK5Hw-MWA H"|$Vy9^B 0 `lA:)ַ:L$sZd)|!5FrI֓YXT'fa.wQw6'b 5Itnعʟz+ҪYc*A#Cy#O&ӹtHc=ЉS3Bla;֙pRLS顁AНL\!dQ9.F19.-M6g_g_C0- {P=#KH X<&KfS^ Y- V6$zZw}^^@Dd8@<  C A2L4|Z9tL'`!L4|Z9tL'v RxQAJAMV f]<yJ0 hUnɂ-$ٻ/|83;;j< 3U݄4 `ԡN^F-'4B,XӶFL]3wCv2UuND"fnBE+_h~Sؒγm 4Iݏq$xٓi 3śELdeN!1F"yLuH-^BfEt%{koK=^'tU, Inpd4 G Ey`7ń`Lk+63S$|׋11ﴦ8M>]hDd( @<  C A2D-NzhzJjȪ`!D-NzhzJjȪ@1 xKA{&"Cp]:i@PH\("I"tVPNv?!_4͏Ŗ8007{3I!A D$1N 1"k(IK&#G3Y!sܵ4Tv?kNX?ܡ84Nk^N;e/n6X>yբdyF8>9H {KM򨼠{rOb1"R ~JWB,+ÇOqDStl;G;MT !)Oj={i7&¥9\7|C xaeR|ХQ&?SQ9P!7l:'elE1t[ dᦻԩ`D+~nv?dQqDd<  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Times New Roman-! := Times New Roman-!p1 Times New Roman-!proc METANative +-LPICTItemName .ObjInfo7e F_979815222ame*:1 F YzTimes New Roman-!( 6!) ^ ! ... eTimes New Roman-!end wTimes New Roman- !rkf45_x :'ࡱ>   ,Times .+  := ( p1)proc)()()) ... )end( :rkf45_xࡱࡱ> T |ࡱ>     > .  Times New Roman-![ !] 3!, !, Symbol-! = Times New RomanPIC68169  FTMETA+ 02<PICTbj 3@ObjInfo-!t Times New Roman-!10 Symbol-! = 8Times New Roman-!( +!) 4!x %Times New Roman-!t 0Times New Roman-!-1.746862466305488 ASymbol-! = Times New Roman-!( !) !y Times New Roman-!t Times New Roman-!.08476007025811474 '@> > >,Times .+ [( 3]( ,),, Symbol(  = ( t) 10)$ = ( +() )( %x) t)-1.746862466305488) = ( () )( y) t).08476007025811474ࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ> ` 2y=-2xy 2 , y0()=1ࡱ>_987068173e6 F"ILOleItemName CompObj7e58 FSObjInfo2ame FOle10Native79dOle10ItemName_979815228e< F`EHPIC0ItemName TAllࡱ> T ࡱ>   +  .f .  Times New Roman-! := 1Times New Roman- ! Exactsoln "-A_META ;=`PICT >ObjInfo_9870681754cA F@m`!Times New Roman-!1 LSymbol-! + 'NTimes New Roman-!x'BTimes New Roman-!2!H!1'W'ࡱ> .f f. .f,Times .+1 := ( Exactsoln"A( L1, Symbol+ + ('Bx(!H2+1ࡱ> ࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ>   x=0.4ࡱ>Allࡱ> Ole0Native    # CompObjmName)*+,./0@C6789S@ObjInfo1HIKJLMNPOQ FbOle10NativejikmlnoprBDwvxyz$Ole10ItemName_979815229ame/OG FiS\ZPIC76007  FTMETA FHTK |ࡱ> K p   .  Times New Roman-![ !] !, Symbol-! = Times New Roman-!x Times New Roman-!.4 Symbol-! = 9Times New Roman-!( *!) 5!y $Times New Roman-!x /Times New Roman-!.8623085097414066 B'|Eࡱ  ,Times .+ [)]( ,, Symbol(PICTNative IObjInfomName_979815231L FÅ`PIC TT24  = ( x).4)# = ( *() )( $y) x).8623085097414066ࡱࡱ> TK |ࡱ> K p   .  Times New Roman-![ !] !METANative KMPICTItemName NObjInfo1 F_979815232JTQ F@FT, Symbol-! = Times New Roman-!x Times New Roman-!.4 Symbol-! = 9Times New Roman-!( *!) 5!y $Times New Roman-!x /Times New Roman-!.8623085097414066 B'|Eࡱ  ,Times .+ [)]( ,, Symbol(  = ( x).4)# = ( *() )( $y) x).8623085097414066ࡱࡱ> TK |ࡱ> PIC0Native TMETAItemName PRPICT8056 S FObjInfo    !"#$%&'()*+,-./012346789:;<=>?BDEFGHIJKLMNOPQRSTUVXYZ[\]^_`aehkmnopqrstuvwxyz{|~K p   .  Times New Roman-![ !] !, Symbol-! = Times New Roman-!x Times New Roman-!.4 Symbol-! = 9Times New Roman-!( *!) 5!y $Times New Roman-!x /Times New Roman-!.8644849248890492 B'|Eࡱ  ,Times .+ [)]( ,, Symbol(  = ( x).4)# = ( *() )( $y) x).8644849248890492Dࡱࡱ> T  ࡱ>     .z .  Times New Roman-! := 2Times New Roman- ! ExactSolnSymbol-! G_979815234V FxPIC+  TMETA UW  PICT XTimes New Roman-!xB "-UsTimes New Roman-!1 `Symbol-! + '\Times New Roman-!1'VTimes New Roman-!x'eTimes New Roman-!2!k'ࡱ> .z z. .z,Times .+2 := ( ExactSoln, Symbol)D (Bx"U( `1('\ + ('V1)x(!k22xsDࡱࡱ> T8T ࡱ> 8    .ObjInfoive_979815235ame.[ F`2PIC68056  FTMETA Z\\  Symbol-! !!!!'!0!9!B!K!T!]!f!o!x!! !!!!'!0!9!B!K!T!]!f!o!x!Times New Roman-!x(k)  ! Exact soln & !h=0.1 e !h=0.01 !h=0.001 Times New Roman-!0!1.:!1.o!1.!1.!.10 !.990100.!1.0o !.991070 !.990200!.2B !.96154B. !.98000Bc !.96330B !.96171B!.3T !.91743T. !.94158Tc !.91969T !.91766T!.4f !.86207f. !.88839fc !.86448f !.86231f!.5x !.80000x. !.82525xc !.80229x !.80023x!.6 !.73529. !.75715c !.73727 !.73549'ࡱ>   , Symbol .+ *~(* * * * * * * * * * * * ( *PICT      ]5"ObjInfo('*km,-./056789@@_979815236GHIJKLMNOP` FUXUX^_`PICdefghijlnoqp wvxy{AT~(* * * * * * * * * * * * ,Times( x(k)) Exact soln)?h=0.1)+h=0.01)/h=0.001(0),1.)51.).1.)21.(0 .1)".99010)A1.)".99107)2.99020(B .2)".96154)5.98000)..96330)2.96171(T .3)".91743)5.94158)..91969)2.91766(f .4)".86207)5.88839)..86448)2.86231(x .5)".80000)5.82525)..80229)2.80023( .6)".73529)5.75715)..73727)2.73549ࡱࡱ> TT ࡱ>  v   .  Symbol-! !!!!'!0!9!B!K!T!]!f!META      _aC"PICT&%('*km,-./0 b56789W@ObjInfoFGHIJKLMNOPUVWXYb`_987068178ghijlnoqpe Fax~o!x!! !!!!'!0!9!B!K!T!]!f!o!x!Times New Roman-!x(k)  !h=0.1 * !h=0.01 X !h=0.001 Times New Roman-!0!05!0f!0!.10 !-.009900& !-.000970W !-.000100!.2B !-.01846B& !-.00176BW !-.00017B!.3T !-.02415T& !-.00226TW !-.00023T!.4f !-.02632f& !-.00241fW !-.00024f!.5x !-.02525x& !-.00229xW !-.00023x!.6 !-.02186& !-.00198W !-.00020'ࡱ  , Symbol .+ *~(* * * * * * * * * * * * ( *~(* * * * * * * * * * * * ,Times( x(k))"h=0.1).h=0.01)0h=0.001(0)'0)10)30(0 .1)-.00990)1-.00097)3-.00010(B .2)-.01846)1-.00176)3-.00017(T .3)-.02415)1-.00226)3-.00023(f .4)-.02632)1-.00241)3-.00024(x .5)-.02525)1-.00229)3-.00023( .6)-.02186)1-.00198)3-.00020ࡱ> ࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ> ` 2y=-4y+4x, y0()Ole0Native cCompObjmNamedgdSObjInfo1e FfOle10Nativeefhgd=1emNameࡱ>Allࡱ> T =4ࡱ>  =   )s .  Symbol-! = Times New Roman-!( !)!Ole10ItemNamei_979815237ame^nk F@ZPIC76007  FjTMETA jllyTimes New Roman-!xSymbol-! - &! + 5Times New Roman-!x! "-/5Times New Roman-!1 /!4"/>D!5 >!4">Times New Roman-!eGTimes New Roman-!( M!) hSymbol-!- RTimes New Roman-!4 YTimes New Roman-!x b'ࡱ> *)s s) )s, Symbol .+ = ,Times( () )(y) x) - ) + (!x"/PICTNative m},ObjInfomName_979815239ep F dPIC0ItemName L*T( /1*4">( >5*4(Ge( M())( R-)4) x ࡱࡱ> TC= 4ࡱ> C= !  ) .  Times New Roman-! := 2METAfoive oqLPICT5235ame r F<ObjInfo6 F_979815240i u F9 a)\Times New Roman- ! ExactSolnSymbol-! GTimes New Roman-!xBSymbol-! - Z! + iTimes New Roman-!xU "-ciTimes New Roman-!1 c!4"crx!5 r!4"rTimes New Roman-!e{Times New Roman-!( !) Symbol-!- Times New Roman-!4 Times New Roman-!x '02ࡱ> <) ) ),Times .+2 := ( ExactSoln, Symbol)D (Bx) - ) + (Ux"c( c1*4"r( r5*4({e( ())( -)4) xࡱ> T$ࡱ>  8   .PIC0ItemName TMETA5229ame tv FxPICT6007 w F8ObjInfo  Symbol-! !!!!'!0!9!B!K!T!]!f!o!x!!!!!! !!!!'!0!9!B!K!T!]!f!o!x!!!!!Times New Roman-!x(k)  ! Exact soln & !h=0.25 c !h=0.5 !h=0.75 Times New Roman-!0!1.:!1.p!1.!1.!.50 !.419180. !.562500d !1.50000 !1.50000!1.0B !.77290B. !.82813Bd!2.B !2.3125B!1.5T !1.2531T. !1.2695Td !2.5000T !9.0625T!2.0f !1.7504f. !1.7549fd!3.f !9.5625f!2.5x !2.2501x. !2.2512xd !3.5000x !12.016x!3.0 !2.7500. !2.7503d!4. !51.578!3.5 !3.2500. !3.2501d !4.5000 !52.078!4.0 !3.7500. !3.7500d!5. !64.785'ࡱ6  , Symbol .+ *(* * * * * * * * * * * * * * * * ( *(* * * * * * * * * * * * * * * * ,Times( x(k)) Exact soln)=h=0.25)1h=0.5)+h=0.75(0),1.)61.).1.).1.(0 .5)".41918)6.56250).1.5000).1.5000(B 1.0)%.77290)6.82813):2.)"2.3125(T 1.5)%1.2531)61.2695).2.5000).9.0625(f 2.0)%1.7504)61.7549):3.)"9.5625(x 2.5)%2.2501)62.2512).3.5000).12.016( 3.0)%2.7500)62.7503):4.)"51.578( 3.5)%3.2500)63.2501).4.5000).52.078( 4.0)%3.7500)63.7500):5.)"64.785ࡱࡱ> ࡱ> _979724397T 55z F`*:AOle@wG 0  wGpHCompObjent``6y|SObjInfol 0 ,@ F Equation Equation Equation9qࡱ> ࡱ> ` dxdt=-2tx 2 , x0()=1 2 ࡱ>Allࡱ> ࡱ>  F Equation Equation EquatiOle10Native{}dOle10ItemName_979724399 F@J`ROle CompObjiveSObjInfo5ame FOle10Native FOle10ItemNameon9qࡱ> ࡱ>  dxdt=5x1-x(), x0()=0.51E~ ࡱ> Allࡱ> ࡱ>  F Equation Equation Equati_979724519e~E F@]clPOle0ItemName CompObj6e FSObjInfomNameon9qࡱ> ࡱ>  2x=-x+xy, x(0)=0.52y=y-xy, y(0)=0.5 &ࡱ> Allࡱ> ࡱ>  F Equation Equation EquatiOle10NativeeOle10ItemName F_979724404 FM}TOle CompObjmNameSObjInfo9ame FOle10Native FOle10ItemNameon9qࡱ> ࡱ>  2x=y,   x(0)=02y=-x+e(1-x 2 )y, y(0)=0.5y, y(0)ࡱ>Allࡱ> ࡱ>  F Equation Equation Equati_979724407ame F෕ߞSOlenfo9ame  FCompObjive FSObjInfomNameon9qࡱ> ࡱ>   eࡱ>Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ> Ole10Native     $"Ole10ItemNamekm,-./056789@_979724409GHIJKLMNOP FII^_`Oledefghijlnoqp wvxy{CompObjiveSObjInfomNameOle10Native FDOle10ItemName F@ e=0.5,1,5,10y 2 ࡱ> Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ>  2x=-(y+z),  x(0)_979724412 F@:`Ole+ CompObjSObjInfoOle10NativeOle10ItemName_987068181 F@Ole15232  F=12y=x+0.2y,  y(0)=12z=0.2-8z+xz, z(0)=10.5ࡱ> Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ> CompObj9 FSObjInfo+Ole10NativeDOle10ItemName %+./04:=ADHKORSTUVW[^beilmnruvz}~@ t n ,X n {}Tiࡱ> Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ>   xt()ࡱ>_987068184T 55 F6 ]Ole@wG 0  wGpHCompObjent``6SObjInfol 0 ,@Ole10Native F$Ole10ItemName_987068186 F&%,VSOlenfo  Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ> ` X N -xt N ()ab<eࡱ>Allࡱ> CompObj9 F SObjInfo Ole10Native dOle10ItemName_987068189e    F=`FqD Ole0ItemName)*+,./0 6789@CompObj6HIKJLMNPOQ FSbObjInfoghjikmlnoprwvxyzࡱ>  F Equation Equation Equation9qࡱ> ࡱ>   en yࡱ>Allࡱ> ࡱ>  F Equation Equation EquatiOle10Native F$Ole10ItemName_987068191 F@W`"_VSOlenfo CompObjiveSObjInfomNameOle10Native FdOle10ItemNameon9qࡱ> ࡱ> ` 2X n -xt n ()abࡱ>Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> _987068194 FyۉOle+ CompObjSObjInfo ࡱ>   2X nࡱ>Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ>   h0ࡱ>Ole10Native!$Ole10ItemName"_987068196 FEOle15232  F#CompObj9 F$SObjInfo+&Ole10Native'$Ole10ItemName(Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ>  a j X n+jj=0k  =hb j f n+jj=0k  , n=0,1,2,KS.doc_987068199e    F௺qD Ole0ItemName)*+,./0 6789)@CompObj6HIKJLMNPOQ F*SbObjInfoghjikmlnoprwvxyz,Ole10Native F-Ole10ItemName1_987068201 FA`iVSOlenfo 22 ࡱ> Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ>   Symbolࡱ>Allࡱ> CompObj9 F3SObjInfo+5Ole10Native6$Ole10ItemName7_987068204e    F@2`qD Ole0ItemName)*+,./0 67898@CompObj6HIKJLMNPOQ F9SbObjInfoghjikmlnoprwvxyz;ࡱ>  F Equation Equation Equation9qࡱ> ࡱ> ` r(q)=a i q ii=0k ࡱ>Allࡱ> ࡱ> Ole10Native F<dOle10ItemName>_987068206 F@VSOlenfo ?CompObjive@SObjInfomNameBOle10Native FCdOle10ItemName FE F Equation Equation Equation9qࡱ> ࡱ> ` s(q)=b i q ii=0k ࡱ>Allࡱ> ࡱ>  F Equation Equation Equati_987068209e    F(.0qD Ole0ItemName)*+,./0 6789F@CompObj6HIKJLMNPOQ FGSbObjInfoghjikmlnoprwvxyzIon9qࡱ> ࡱ> @ r(1)=0, s(1)="r(1)fࡱ> Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> Ole10NativeJDOle10ItemNameL_987068212 FJZOle15232  FMCompObjive FNSObjInfomNamePOle10Native FQOle10ItemNameXࡱ>  Y n+1 =Y n +hc, r k rr=1p  k 1 =ft n ,X n ()k r =ft n +ha r ,X n +hb rs k ss=1p  (), r=2,3,Kpࡱ> Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ> @ rq()=q-12 erW 2 ࡱ> Allࡱ> _987068214e    FQs@y|qD Ole0ItemName)*+,./0 6789Y@CompObj6HIKJLMNPOQ FZSbObjInfoghjikmlnoprwvxyz\Ole10Native]DOle10ItemName__987068216 F  Ole `ࡱ>  F Equation Equation Equation9qࡱ> ࡱ> ` r(1)=0, c rr=1m  ="r(1)ࡱ>Allࡱ> ࡱ> CompObjive FaSObjInfomNamecOle10Native FddOle10ItemNamef_987068219T 55 F=Ole@wG 0  wGpHgCompObjent``6hSObjInfol 0 j,@ F Equation Equation Equation9qࡱ> ࡱ>  (a) r(q)=q-1  c rr=14  =16+13+13+16=1 "r(q)=1ࡱ>Allࡱ> Ole10NativekOle10ItemNameo_987068222 FOle pࡱ>  F Equation Equation Equation9qࡱ> ࡱ>  r(1)=0 "r(1)=1=c rr=14    1 ࡱ> Allࡱ> CompObjive FqSObjInfomNamesOle10Native FtOle10ItemNamew_987068224e  F@`a qSOlenfomName xCompObjive   FySObjInfomName{ࡱ>  F Equation Equation Equation9qࡱ> ࡱ>  (b) r(q)=q 2 -q  s(q)=32q-12 "r(q)=2q-1ࡱ>Allࡱ> Ole10Native   F|Ole10ItemName_987068227 F@*R$VSOlenfo ࡱ>  F Equation Equation Equation9qࡱ> ࡱ> ` r(1)=0 "r(1)=1=s(1)2pࡱ>Allࡱ> ࡱ> CompObjiveSObjInfomNameOle10Native FdOle10ItemName_987068229e F<<SOlenfomName CompObjive FSObjInfomName F Equation Equation Equation9qࡱ> ࡱ>   x n3fࡱ>Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> Ole10NativeT 55 F$Ole10ItemName 0 wGpH_987068232t``6& FVM^hSOlenfol 0 ,@CompObjiveSObjInfomNameOle10Native FDOle10ItemNameࡱ> @ 1"r1()h n+%2ࡱ> Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ>       !"#$%&'()*+,-./01356798;:<=?>@BACEDFHGIJKNLMPOQRUSV}WXZY[\]_^`abcdfeghikjlmnpoqsrtuvxwy{z|~`q0xAKAmMlCh}Z+d$MiHIyY'!&qĸI~7֚*Js]rL.[VmN&ʋ7Y JC>5%]ozuuqV֡Qh[٭-a_yyru@o[%Bx"߃BxoK=?'Qfqq]Lbb[w3YK3HDd4 < 2 C A212>l3 5BcB)`!l3 5BcB XxS=KAmUY_2u +@]cj >@FXOjqp?bECN%qECFm1yT -wpsDd2M < 5 C A54be% q[9AۍAn9% q[9AۍPNG  IHDR2gAMAIDATxі8P2- 8O5V #ܠH%K#G,!XB8b q8V)e+`g˾SB8b 5%wLvb 4H=4H "%K#G,! 8X#G,!XB8b q-!%QB"||.SX_#*d+upX$NDk*˩i!` #G,!Ίk˾-',r=Q⸃8b q%KAq4H "k+]黻 Ē;軾tK#G,!N-/ǡ5 8b q4ڻpg˛c(b! %QBuELunV_["8+bYJqlȺXBE+X,ÔY7n;ז}OMmq iuw}8b 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FSObjInfo     "Ole10Native*km,-./056789$@Ole10ItemNameIJKLMNOPUVWXY`_987068414ghijlnoqp F~ࡱ>   R A2 ࡱ>Allࡱ> ࡱ>  F Equation Equation Equation9qࡱ> ࡱ>  z=re if ()sOle0ItemName CompObj1ame FSObjInfo7 FOle10Nativee if (), 0f<2p 2 ࡱ> Allࡱ> ࡱ>  FMathType Equation Equation Equation9qࡱ> ࡱ>  X n+1 = 13 4XOle10ItemName_987068671ame FuuDOle75907  FCompObj\ObjInfomNameOle10Nativee FOle10ItemName F_987076661h  F@`qS n -X n-1 {}+2h3ft n+1 ,X n+1 ()hTIhTIࡱAllࡱ> ࡱ>  FMathType Equation Equation Equation9qࡱ> ࡱ> Ole0ItemNameyInformation CompObj1ame F\ObjInfo6 FOle10Native F p(q;hl)q 2 - 43 q+ 13 - 23 hlq 2 =0o Equation in C:\Dynࡱ> Allࡱ> ࡱ>  FMathType Equation Equation Equation9qࡱ> Ole10ItemName F_987076764ame F n nkOleObjive  FCompObjmName\ObjInfo8eT 55 FOle10Nativee 0 wGpHD_987076514e``6 F'0\OlenfomName 0 ,@ࡱ> @ z=hlpe=%2ࡱ> ࡱ>  FMathType Equation Equation Equation9qࡱ> ࡱ>  z= 12 3q 2 -4CompObj8eT 55 F\ObjInfomName 0 wGpHOle10Native``6 FOle10ItemName 0 ,@q+1q 29|D|Dࡱ> Allࡱ> ࡱ>  FMathType Equation Equation Equation9qࡱ> ࡱ> ` q=e ifP&Yess%2_987076985      F "Ole&%('*km,-./0 56789@CompObjFGHIJKLMNOPUVWXY\`ObjInfofghijlnoqpwvxy{Ole10Native Fd_987077066ame  F`Nc`NckOleObjive  FCompObjmName  \P&Nࡱ>ࡱ>  FMathType Equation Equation Equation9qࡱ> ࡱ>  z= 12 3e 2if -4e if +1e 2if = 32 -2e -ij + 12 eObjInfo8eT 55 FOle10Nativee 0  wGpHOle10ItemName`6 F_987077503ame 0  F #,@ -2ifࡱ> Allࡱ> ࡱ>  FMathType Equation Equation Equation9qࡱ> ࡱ> @ R A=%2Ole0ItemName CompObj1ame F\ObjInfo7 FOle10NativeD./ࡱ> ࡱ>  FMathType Equation Equation Equation9qࡱ> ࡱ> @ z=1hType=%2ࡱ> ࡱ> _987077559ame< F JqOle68346ee  FCompObjmName F\ObjInfo42 FOle10Native   #D _987077607'()*+,./0 F۾?B@OleEFGHIKJLMNPOQ UWXY\bCompObjghjikmlnoprwvxyz\ FMathType Equation Equation Equation9qࡱ> ࡱ> @ p(q;hl)2ࡱ> Allࡱ> ࡱ>  FMathType Equation Equation ObjInfomName FOle10Nativee FDOle10ItemName F_987077638ame! 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