Need gradual data values.

This is the data values I got for my equation. It is not smooth curve. If the data values are gradual, the curve is smooth. Generally, Where I rectify to get gradual values?

0.0320415867172945

0.184398415806668

0.0330722761271637

0.00593158809660344

0.00106384384348042

0.000190802817875919

3.42209202342026e-05

6.13760003501216e-06

1.10079255414445e-06

1.97429653341276e-07

3.54094582777690e-08

6.35076704185715e-09

1.13902454263924e-09

2.04286647610229e-10

3.66392758273023e-11

6.57133762217517e-12

1.17858437890949e-12

2.11381794403936e-13

3.79118065749105e-14

6.79956891191319e-15

1.21951818087333e-15

2.18723365076115e-16

3.92285340067338e-17

7.03572697769140e-18

1.26187366818545e-18

2.26319918255593e-19

4.05909931323577e-20

7.28008712697735e-21

1.30570021786759e-21

2.34180309823753e-22

4.20007722742912e-23

7.53293422903285e-24

1.35104892187110e-24

2.42313703238507e-25

4.34595149195947e-26

7.79456304700777e-27

1.39797264663868e-27

2.50729580216841e-28

4.49689216358189e-29

8.06527858156800e-30

1.44652609473487e-30

2.59437751788373e-31

4.65307520534857e-32

8.34539642645792e-33

1.49676586861698e-33

2.68448369732820e-34

4.81468269174236e-35

8.63524313641209e-36

1.54875053662087e-36

2.77771938414703e-37

4.98190302093494e-38

8.93515660784508e-39

1.60254070123860e-39

2.87419327029082e-40

5.15493113441686e-41

9.24548647276300e-42

1.65819906976727e-42

2.97401782272611e-43

5.33396874425575e-44

9.56659450635627e-45

1.71579052741179e-45

3.07730941454665e-46

5.51922456824746e-47

9.89885504874850e-48

1.77538221292648e-48

3.18418846063841e-49

5.71091457323429e-50

1.02426554413933e-50

1.83704359688380e-51

3.29477955805639e-52

5.90926222687402e-53

1.05983964786276e-53

1.90084656265939e-54

3.40921163115544e-55

6.11449875137923e-56

1.09664924972184e-56

1.96686338438106e-57

3.52750066766359e-58

6.32031682314549e-59

1.09823600535655e-59

Thanks for advance.

[Torsten]

Evaluate your equation for more points in between x=1 and x=3 (or whatever the independent variable for your equation is).

[Nathi]

clc; close all; clear all;

ymax=20; m=80; dy=ymax/m; %y=dy:dy:ymax;        %'i'th row
xmax=1; n=20; dx=xmax/n;                        %'j'th column   
tmax=100; nt=500; dt=tmax/nt; t=0:dt:tmax;     
UOLD=zeros(m,nt); VOLD=zeros(m,nt); 
TNEW=0; TOLD=TNEW*ones(m,nt); TWALL=ones(1,length(t));
A=zeros([1,m]); 
B=A;
C=A;
D=A;
T=TOLD; 
tic
    for j=1:nt
        for i=1:m
            if j>i
                C(i)=(dt*VOLD(i,j)/4*dy)-(dt/(2*dy^2));
            elseif i>j
                A(i)=(-dt*VOLD(i,j)/4*dy)-(dt/(2*dy^2));
            elseif i==j
                B(i)=1+(dt*UOLD(i,j)/2*dx)+(dt/(dy^2));
            end
        end
    end
    for j=2:nt
        if j==1
            for i=2:m-1
                if i==2
                    D(i)=(-dt*UOLD(i,j)*(-TNEW+TOLD(i,j)-TNEW)/2*dx)+(dt/(2*dy^2)*(TWALL(j)-2*TOLD(i,j)+TOLD(i-1,j)))-(dt*VOLD(i,j)/4*dy*(TOLD(i-1,j)-TWALL(j)+TOLD(i,j)))-(-dt/4*dy*VOLD(i,j))-(dt/2*dy^2*TWALL);
                elseif i==m-1
                    D(i)=(-dt*UOLD(i,j)*(-TNEW+TOLD(i,j)-TNEW)/2*dx)+(dt/(2*dy^2)*(TOLD(i+1,j)-2*TOLD(i,j)+TNEW))-(dt*VOLD(i,j)/4*dy*(TNEW-TOLD(i+1,j)-TOLD(i,j)))-(dt/4*dy*VOLD(i,j))-(dt/2*dy^2*TNEW);
                else
                    D(i)=(-dt*UOLD(i,j)*(-TNEW+TOLD(i,j)-TNEW)/2*dx)+(dt/(2*dy^2)*(TOLD(i+1,j)-2*TOLD(i,j)+TOLD(i-1,j)))-(dt*VOLD(i,j)/4*dy*(TOLD(i-1,j)-TOLD(i+1,j)+TOLD(i,j)));
                end
            end
        else
            for i=2:m-1
                if i==2
                     D(i)=(-dt*UOLD(i,j)*(-T(i,j-1)+TOLD(i,j)-TOLD(i,j-1))/2*dx)+(dt/(2*dy^2)*(TWALL(j)-2*TOLD(i,j)+TOLD(i-1,j)))-(dt*VOLD(i,j)/4*dy*(TOLD(i-1,j)-TWALL(j)+TOLD(i,j)))-(-dt/4*dy*VOLD(i,j))-(dt/2*dy^2*TWALL(j));
                elseif i==m-1
                    D(i)=(-dt*UOLD(i,j)*(-T(i,j-1)+TOLD(i,j)-TOLD(i,j-1))/2*dx)+(dt/(2*dy^2)*(TOLD(i+1,j)-2*TOLD(i,j)+TNEW))-(dt*VOLD(i,j)/4*dy*(TNEW-TOLD(i+1,j)+TOLD(i,j)))-(dt/4*dy*VOLD(i,j))-(dt/2*dy^2*TNEW);
                else
                    D(i)=(-dt*UOLD(i,j)*(-T(i,j-1)+TOLD(i,j)-TOLD(i,j-1))/2*dx)+(dt/(2*dy^2)*(TOLD(i+1,j)-2*TOLD(i,j)+TOLD(i-1,j)))-(dt*VOLD(i,j)/4*dy*(TOLD(i-1,j)-TOLD(i+1,j)+TOLD(i,j)));
                end
            end
        end
        T(:,j)=TriDiag(A,B,C,D);
        dt=0.2+dt;
        TOLD=T;
end
 

@Torsten As you said to evaluate the first three points in x(i.e., I'm taking i =1:m) only the matters now. but i couldn't identity the position in my equation and how to rectify the first three points to obtain gradual values ?.

eventhough, I'm using iteration for better smooth curve, but the first three points only stuck in my case.

please resolve my issue. thank you

Answer 1

[John D'Errico]

Actually, it IS quite a smooth curve. Almost perfectly smooth, in fact.

y = [0.0320415867172945
    0.184398415806668
    0.0330722761271637
    0.00593158809660344
    0.00106384384348042
    0.000190802817875919
    3.42209202342026e-05
    6.13760003501216e-06
    1.10079255414445e-06
    1.97429653341276e-07
    3.54094582777690e-08
    6.35076704185715e-09
    1.13902454263924e-09
    2.04286647610229e-10
    3.66392758273023e-11
    6.57133762217517e-12
    1.17858437890949e-12
    2.11381794403936e-13
    3.79118065749105e-14
    6.79956891191319e-15
    1.21951818087333e-15
    2.18723365076115e-16
    3.92285340067338e-17
    7.03572697769140e-18
    1.26187366818545e-18
    2.26319918255593e-19
    4.05909931323577e-20
    7.28008712697735e-21
    1.30570021786759e-21
    2.34180309823753e-22
    4.20007722742912e-23
    7.53293422903285e-24
    1.35104892187110e-24
    2.42313703238507e-25
    4.34595149195947e-26
    7.79456304700777e-27
    1.39797264663868e-27
    2.50729580216841e-28
    4.49689216358189e-29
    8.06527858156800e-30
    1.44652609473487e-30
    2.59437751788373e-31
    4.65307520534857e-32
    8.34539642645792e-33
    1.49676586861698e-33
    2.68448369732820e-34
    4.81468269174236e-35
    8.63524313641209e-36
    1.54875053662087e-36
    2.77771938414703e-37
    4.98190302093494e-38
    8.93515660784508e-39
    1.60254070123860e-39
    2.87419327029082e-40
    5.15493113441686e-41
    9.24548647276300e-42
    1.65819906976727e-42
    2.97401782272611e-43
    5.33396874425575e-44
    9.56659450635627e-45
    1.71579052741179e-45
    3.07730941454665e-46
    5.51922456824746e-47
    9.89885504874850e-48
    1.77538221292648e-48
    3.18418846063841e-49
    5.71091457323429e-50
    1.02426554413933e-50
    1.83704359688380e-51
    3.29477955805639e-52
    5.90926222687402e-53
    1.05983964786276e-53
    1.90084656265939e-54
    3.40921163115544e-55
    6.11449875137923e-56
    1.09664924972184e-56
    1.96686338438106e-57
    3.52750066766359e-58
    6.32031682314549e-59
    1.09823600535655e-59];
x = 0:numel(y)-1;
semilogy(x,y,'o')

So only the very first point is at all in error. I would note that the first data point seems to be in error. But we can fix that. We can see the problem in another way from here:

diff(log(y))

ans = 79×1

1.7501 -1.7184 -1.7184 -1.7184 -1.7184 -1.7184 -1.7184 -1.7184 -1.7184 -1.7184

And that plot suggests there is no need at all to smooth the curve. Ok, if you prefer, the first data point is just a bit off. We can use a simple straight line fit against log(y) to correct the first point.

P1 = polyfit(x(2:end),log(y(2:end)),1)

P1 = 1×2

-1.7184 0.0286

So the very first data point SHOULD have been approximately:

exp(P1(2))

ans = 1.0290

semilogy(x,y,'bo',0,exp(P1(2)),'rs')
grid on

You should see that the red square now appears to be perfectly in line with the rest of the curve.

Again, all that was needed was to look at your curve in the correct way.

[Nathi]

@John D'Errico You exactly got my point. Yes, it's first three points only error and it's only not gradually increasing and decreasing and all other points are perfectly fit well. I'm using this technique and it's not quite acceptable.

For example,

1 1.7813 2.339 2.6461 2.7189 2.603 2.3574 2.0404 1.7003 1.371 1.0761 0.8234 0.6161 0.4516 0.3248 0.2294 0.1592 0.1087 0.0731 0.0483 0.0315 0.0202 0.0128 0.008 0.0049 0.003 0.0018 0.001 0.0006 0.0003 0.0002 0.0001 0.0001

Please plot this data. the curve comes well. but the above one edges are sharp. but i don't want the sharp edges. it should be gradual values like this data.