ME201/MTH281ME400/CHE400 Convergence of Bessel Expansions

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Anastasia Townsend
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1 ME201/MTH281ME400/CHE400 Convergence of Bessel Expansions 1. Introduction This notebook provides a general framework for the construction and visualization of partial sums of Fourier-Bessel series. The particular Bessel eigenfunctions and eigenvalues are defined for Mathematica in section 2. The specific function to be expanded, and its Fourier-Bessel coefficients are defined in section 3. The example used here is the constant function. Section 4 contains the definition of the terms and partial sums of the series. Section 5 defines graphs of any given partial sum. Section 6 defines a computationally efficient routine for producing a sequence of all partial sums up to a specified value n. Section 7 shows how to send the graph sequence to a Manipulate panel for convenient visualization. Section 8 gives a summary by applying the routines to a new example. 2. Specification of Fourier-Bessel Basis Functions In the first part of this notebook, we consider expansions in the Bessel functions J 0 on an interval [0,a]. These functions are the eigenfunctions of the Sturm-Liouville problem given below. d dr r dy dr + lry = 0, 0 < r < a, with y well behaved at r = 0, and y HaL = 0. As we showed in class, the nth eigenfunction is y n HrL = J 0 Ha n rêal, where a n is the nth root of J 0. These eigenfunctions are orthogonal on [0, a] with respect to the weight function r. Any piecewise smooth function f(r) on [0, a] can be expanded in these eigenfunctions. The coefficients in the expansion are calculated using orthogonality. The result is f(r) = n=1 C n y n HrL, where C n = Ÿ a 0 f HrL yn HrL r r a Ÿ 8yn 0 HrL< 2. r r We define for Mathematica the eigenfunctions, the normalization integrals, the eigenvalues, the interval endpoint a, and nmax, the maximum number of terms to be used in any expansion. In[1]:= a = 3; In[2]:= nmax = 101; Now we get the first nmax zeros of J0 and store them in the list beszer. In[3]:= beszer = N[Table[BesselJZero[0,i],{i,1,nmax}]]; We construct a list of the values of a n êa, which will appear in the arguments of the Bessel functions defining the eigenfunctions, and store these in the list eiger. In[4]:= eiger = beszer/a; Now we can define the eigenfunctions. In[5]:= y[r_,n_] := BesselJ[0,r*eiger[[n]]]

2 2 genbessexp.nb Finally we define the nth normalization integral norm[n], and make a list, normlist, of the first nmax values. Although in this case the normalization integral can be found analytically, with more general boundary conditions that may not be so. For that reason we use a numerical integration to calculate the normalization integrals. In[6]:= In[7]:= norm@n_d := NIntegrateAHy@r, ndl 2 r, 8r, 0, a<, AccuracyGoal Ø 8E normlist = Table@norm@nD, 8n, 1, nmax<d; 3. Definition of Function and Expansion Coefficients The function to be expanded is a constant function, defined below for Mathematica. The interval is [0,a], where a has already been assigned a value in section 2. For graphics purposes, we assign a value to the variable pltrange. This value will be picked up by our graphics routines and used for the PlotRange setting. The maximum of this function on [0, a] is 1, so we choose a range [-0.3, 1.3]. This is large enough to catch the overshoot in the earlier partial sums. In[8]:= f[r_] := In[9]:= pltrange = 8-0.3, 1.3<; We now define a routine calcoeff which, when executed, will calculate the Fourier-Bessel coefficients numerically. It will use the values of the normalization integral calculated in section 2, and it will calculate numerically the integral Ÿ 0 a f HrL yn r r. The coefficient values are stored in the function coeff. In[10]:= calcoeff := Module@8n<, Clear@coeffD; Do@coeff@nD = H1 ê normlist@@nddl NIntegrate@ y@r, nd r, 8r, 0, a<, AccuracyGoal Ø 8D, 8n, 1, nmax<dd Now we calculate the coefficients by executing calcoeff. In[11]:= calcoeff For this particular function, the coefficients are also known analytically. We use those known values to check the accuracy of our numerical work. As we showed in class, the nth coefficient in this expansion is given by 2/a n J 1 (a n ). We define this for Mathematica. In[12]:= analytcoeff@n_d := 2.0 ê Hbeszer@@nDD BesselJ@1, beszer@@ndddl We calculate the difference between the numerical and analytical coefficients.

3 genbessexp.nb 3 In[13]:= Table@coeff@nD - analytcoeff@nd, 8n, 1, nmax<d Out[13]= µ 10-15, µ 10-15, µ 10-14, µ 10-14, µ 10-14, µ 10-14, µ 10-14, µ 10-14, µ 10-15, µ 10-14, µ 10-15, µ 10-14, µ 10-14, µ 10-15, µ 10-15, µ 10-14, µ 10-15, µ 10-14, µ 10-16, µ 10-16, µ 10-15, µ 10-14, µ 10-14, µ 10-15, µ 10-14, µ 10-15, µ 10-15, µ 10-15, µ 10-15, µ 10-16, µ 10-16, µ 10-15, µ 10-15, µ 10-15, µ 10-14, µ 10-15, µ 10-15, µ 10-15, µ 10-15, µ 10-15, µ 10-15, µ 10-15, µ 10-15, µ 10-15, µ 10-16, µ 10-15, µ 10-15, µ 10-15, µ 10-14, µ 10-15, µ 10-15, µ 10-15, µ 10-15, µ 10-15, µ 10-15, µ 10-15, µ 10-15, µ 10-15, µ 10-14, µ 10-15, µ 10-15, µ 10-16, µ 10-14, µ 10-15, µ 10-12, µ 10-12, µ 10-12, µ 10-12, µ 10-15, µ 10-15, µ 10-16, µ 10-15, µ 10-15, µ 10-15, µ 10-15, µ 10-15, µ 10-15, µ 10-12, µ 10-14, µ 10-15, µ 10-15, µ 10-14, µ 10-14, µ 10-14, µ 10-15, µ 10-14, µ 10-14, µ 10-14, µ 10-14, µ 10-15, µ 10-15, µ 10-14, µ 10-15, µ 10-15, µ 10-15, -103 µ 10-14, µ 10-15, µ 10-15, µ 10-15, µ 10-15, µ = The agreement is excellent. In the remainder of this notebook we will use the coefficients defined numerically. If you wish to use analytical expressions for the expansion coefficients in any problem, simply load them into the function coeff, and do not execute calcoeff. 4. Terms and Partial Sums of Fourier-Bessel Series Now we define the nth term of the series, called fourterm[r,n], and the nth partial sum foursum[r,n]. In[14]:= In[15]:= fourterm[r_,n_] := coeff[n]*y[r,n] foursum[r_,n_] := Sum[fourterm[r,k],{k,1,n}] 5. Graphs of f[r] and Partial Sums of Fourier-Bessel Series There are two graphical functions defined here. The function pic[n] gives a graph of the function f[r] and of the nth partial sum of the Fourier-Bessel series. The function picfunc gives a plot of f[r] only. The function f is in blue, and the Fourier-Bessel series is in red. The range that is plotted is user-specified. It is assigned to the variable pltrange, which we did in section 3. The number of points plotted in each graph is specified by the variable npoints. The default, set below, is 500. If your computation times seem excessive, you can make this number smaller. The graph size is set by the SetOptions command below. The value chosen for ImageSize, namely 250, is appropriate for printed graphs. For computer display a larger value might be better. In[16]:= SetOptions@Plot, ImageSize -> 250D; In[17]:= npoints = 500; In[18]:= picfunc := Plot[f[r],{r,0,a},PlotStyle -> {RGBColor[0,0,1],Thickness[0.004]},PlotPoints -> npoints, PlotRange -> pltrange, AxesLabel -> {"r","f[r]"},aspectratio -> 0.7]

4 4 genbessexp.nb In[20]:= pic[n_] := Plot[{foursum[r,n],f[r]},{r,0,a}, PlotStyle -> {{RGBColor[1,0,0],Thickness[0.004]},{RGBColor[0,0,1],Thickness[0.004]}}, PlotPoints -> npoints,plotrange -> pltrange, AxesLabel -> {"r","f[r]"},aspectratio -> 0.7, PlotLabel -> Row[{"n = ",PaddedForm[n,2]}]] Let's try this out. We produce first a graph of the basic function, then a graph of the partial sums for n = 6, and then a sequence of graphs of partial sums up to and including n = 6. We produce the sequence by a Table command, and we use the command GraphicsGrid to print two graphs per line. In[21]:= picfunc Out[21]= r In[22]:= pic[6] n = 6 Out[22]= r

5 genbessexp.nb 5 In[23]:= GraphicsGrid@Table@8pic@iD, pic@i + 1D<, 8i, 1, 5, 2<DD n = 1 n = r r n = 3 n = 4 Out[23]= r r n = 5 n = r r It is clear that many more terms are needed to get a result closely resembling the original f[r]. It is possible to produce a much longer sequence of graphs by this same technique, but the computation time becomes large. The problem is that our technique is very inefficient. For each graph in the sequence, we are recomputing all of the previous terms. An efficient technique would store the result for partial sum k and use it to compute partial sum k+1. We develop such a technique next, and use it to generate a large graph sequence.

6 6 genbessexp.nb 6. Efficient Production of a Sequence of Partial Sum Graphs Our technique is to find and save the values of the kth partial sums at every plotted point. Then to get the partial sums for k+1, we only have to increment those values by the value of the new term at each point. Thus each succeeding graph requires the evaluation of only one term in the series. However, we must then change our graphing technique, because Plot works only for functions defined analytically. For the present case, we can use ListPlot, which plots a given numerical set of points. The routine defined here is called picarray[first, last, grinc], where all arguments are integers. The argument "first" is the n value of the first partial sum in the sequence and the argument "last" is the last n value. The argument "grinc" specifies the step between displayed graphs. All the partial sums are calculated, but by choosing grinc greater than 1, you can display every "grincth" graph. The first four functions defined below are used by picarray to calculate coordinate lists and to produce graphs. In[24]:= SetOptions@ListPlot, ImageSize -> 250D; In[25]:= In[26]:= In[27]:= In[28]:= mksumlist[n_] := Module[{ans,r,inc,j}, ans = {{0,foursum[0,n]}}; inc = (a)/npoints; Do[r = j*inc; ans = Append[ans,{r,foursum[r,n]}],{j,1,npoints}]; ans] mktermlist[n_] := Module[{ans,r,inc,j}, ans = {{0.0,fourterm[0,n]}}; inc = a/npoints; Do[r = j*inc; ans = Append[ans,{0.0,fourterm[r,n]}],{j,1,npoints}]; ans] funclist := Module[{ans,r,inc,j}, ans = {{0,f[0]}}; inc = a/npoints; Do[r = j*inc; ans = Append[ans,{r,f[r]}],{j,1,npoints}]; ans] mkgraph@list_, rcol_, gcol_, bcol_d := ListPlot@list, Joined Ø True, PlotStyle Ø 8RGBColor@rcol, gcol, bcold, Thickness@0.004D<, PlotRange Ø pltranged In[29]:= picarray[first_,last_,grinc_] := Module[ {sumlist,termlist,k,grph,grph0}, sumlist = mksumlist[first]; grph0 = mkgraph[funclist,0,0,1]; grph = mkgraph[sumlist,1,0,0]; Print[Show[{grph0,grph},AxesLabel -> {"r","f[r]"}, AspectRatio -> 0.7, PlotLabel -> Row[{"n = ",PaddedForm[first,2]}]]]; Do[sumlist = sumlist+mktermlist[k]; If[(Mod[k-first,grinc]== 0), (grph = mkgraph[sumlist,1,0,0]; Print[Show[{grph0,grph},AxesLabel -> {"r","f[r]"}, AspectRatio -> 0.7, PlotLabel -> Row[{"n =",PaddedForm[k,2]}]]])], {k,first+1,last}]] As an example, we execute picarray[1,5,4]. This will start with n = 1 and then display every 4th graph up to n = 5 -- i.e., graphs 1 and 5. In[30]:= picarray[1,5,4]

7 genbessexp.nb 7 n = r n = r Now we use this new faster method to display the partial sums for our function up to and including n = 101. We display every graph in the sequence by setting grinc = 1. In the printed version of the notebook, the graphs of the sequence are combined into a single cell, so only the first graph is visible. To animate the graph sequence, select the cell containing the graphs and use the menu item Graphics->Rendering->Animate Selected Graphics. In[31]:= picarray[1,101,1]; n = r For visualization in the printed version of this notebook, we show every 20th graph in the sequence. In[32]:= picarray@1, 101, 20D;

8 8 genbessexp.nb n = r n = r n = r n = r

9 genbessexp.nb 9 n = r n = r 7. Using Manipulate to Visualize the Graph Sequences We saw in section 6 above that many lines of Mathematica code were required to define a function which would produce efficiently a graph sequence of partial sums. Here we accomplish the same thing with a single command Manipulate, although to be honest we make use here also of the many lines of code defined earlier in our use of Manipulate. The power of manipulate is only evident in a fully interactive mode, so you need to execute this Mathematica notebook to appreciate what Manipulate can do. We start by modifying the earlier code for picarray[first, last, grinc] to produce a Manipulate panel as an output, rather than a sequence of printed graphs. For convenience, we repeat the definition of the arguments of picarray here. The argument first is the n value of the first partial sum in the sequence and the argument last is the last n value. The variable grinc specifies the step between displayed graphs. All the partial sums are calculated, but by choosing grinc greater than 1, you can display every "grincth" graph. Now we define a new command manpicarray[first,last,grinc]. It does exactly what picarray does except that the output is now displayed in a Manipulate panel. In[33]:= manpicarray[first_,last_,grinc_] := DynamicModule[ {sumlist,termlist,k,grph,grph0,mangraph}, sumlist = mksumlist[first]; grph0 = picfunc; grph = mkgraph[sumlist,1,0,0]; mangraph[first] = Show[{grph0,grph},AxesLabel -> {"r","f[r]"}, AspectRatio -> 0.7, PlotLabel -> Row[{"n =",PaddedForm[first,2]}]]; Do[sumlist = sumlist+mktermlist[k]; If[(Mod[k-first,grinc]== 0), (grph = mkgraph[sumlist,1,0,0]; mangraph[k] = Show[{grph0,grph},AxesLabel -> {"r","f[r]"}, AspectRatio -> 0.7, PlotLabel -> Row[{"n =",PaddedForm[k,2]}]])], {k,first+1,last}]; Manipulate[mangraph[n],{n,first,last,grinc}]]

10 10 genbessexp.nb We now use this to construct the 101 graph sequence of the previous section, but now assigning it to a Manipulate panel. In[34]:= manpicarray@1, 101, 1D n 11 n = 11 Out[34]= r By experimenting with the slider and the movie controls, we can study the convergence of this series. It is what we would have called a 1/n type of convergence in our Fourier series, and in fact with some labor one can show that the large terms in the series do drop off like 1/n. There is a Gibbs peak near r = 3. There is also some serious oscillation at r = 0, but unlike the persistent Gibbs peak at r = 3, the oscillation at r = 0 eventually dies down. 8. Summary We summarize the calculations of this notebook by carrying out another example. The eigenfunctions we use are J 1 ' s. They are eigenfunctions of the Sturm-Liouville system specified below. d dr r dy dr + l r - 1 r y = 0, 0 < r < a, with y well behaved at r = 0, and y HaL = 0. The nth eigenfunction is y n HrL = J 1 Hb n rêal, where b n is the nth root of J 1. These eigenfunctions are orthogonal on [0, a] with respect to the weight function r. Any piecewise smooth function f(r) on [0, a] can be expanded in these eigenfunctions. The coefficients in the expansion are calculated using orthogonality. The result is f(r) = n=1 C n y n HrL, where C n = Ÿ a 0 f HrL yn HrL r r a Ÿ 8yn 0 HrL< 2. r r We make the basic settings of interval length (taken as 2 here), maximum number of terms, the list of zeros beszer, the square roots of the eigenvalues eiger, and the eigenfunctions y. We also calculate the normalization integrals. In[35]:= a = 2; In[36]:= nmax = 101; In[37]:= beszer = N[Table[BesselJZero[1,i],{i,1,nmax}]];

11 genbessexp.nb 11 Now we construct a list with the square roots of the eigenvalues -- ie., the values of a n êa. In[38]:= eiger = beszer/a; Then we can define the eigenfunctions. In[39]:= y[r_,n_] := BesselJ[1,r*eiger[[n]]] We calculate and store the first nmax normalization integrals. In[40]:= In[41]:= norm@n_d := NIntegrateAHy@r, ndl 2 r, 8r, 0, a<, AccuracyGoal Ø 8E normlist = Table@norm@nD, 8n, 1, nmax<d; Now we specify the function to be expanded, and calculate the Fourier-Bessel coefficients. In[42]:= f[r_] := r In[43]:= pltrange = 8-0.3, 2.3<; In[44]:= calcoeff We construct the graph sequence for partial sums up to 101, and send the graphs to a manipulate panel with the command manpicarray. In[45]:= manpicarray@1, 101, 1D n n = Out[45]= r Exploring with the slider and the movie controls, we see that the result is similar to our first example. The Gibbs phenomenon is showing up at the endpoint r = 2, and there is some oscillation around r = 0. Note that only about 10 lines of code were necessary to produce this interactive display of the Fourier-Bessel series.

ME 201/MTH 281/ME400/CHE400 Bessel Functions

ME 201/MTH 281/ME400/CHE400 Bessel Functions ME 01/MTH 81/ME00/CHE00 Bessel Functions 1. Introduction This notebook has two purposes: to give a brief introduction to Bessel functions, and to illustrate how Mathematica can be used in working with