In this section, we present numerical examples obtained by using Matlab language [70,71]
to illustrate the usefulness of the proposed method. Since the analytic solution of (1) is difficult to obtain, we construct the final datag(r)by solving the forward problem with the given dataϕ(r)andψ(r)by the finite difference method.
We chooseT =1,r0=π. Lett=T/Nandr=π/Mbe the step sizes for time and space variables, respectively. The grid points in the time interval are labelledtk=kt, k=1, 2,. . .,N, the grid points in the space interval areri=ir,i=1, 2,. . .,M, and set uki =u(ri,tk).
Based on the idea in [61–63,66–69], we approximate the time-fractional derivatives by
Dαtu(ri,tk)= (t)1−α (3−α)
⎡
⎣b0 1
t(uki −uk−1i )
−
k−1
j=1
(bk−j−1−bk−j) 1
t(uji−uji−1)−bk−1ν(xi)
⎤
⎦, (96)
wherei=1, 2,. . .,M−1,k=1, 2,. . .,Nandbj =(j+1)1−α−j1−α. We approximate the space derivatives by
ur(ri,tk)≈ uki+1−uki
r , (97)
urr(ri,tk)≈ uki+1−2uki +uki−1
(r)2 . (98)
Next, the measured data is given by the following random form
fδ(r)=f(r)+δãf(r)(2rand(size(f(r)))−1), (99) gδ(r)=g(r)+δãg(r)(2rand(size(g(r)))−1). (100) In order to make the sensitivity analysis for numerical results, we calculate theL2[0,r0;r2] error by
e(ϕ)=
# 1 M+1
(ϕ(r)−ϕmi,δ(r))2, i=1,. . ., 6. (101)
The relativeL2[0,r0;r2] error is defined by eR(ϕ)=
$(ϕ(r)−ϕmi,δ(r))2
$ϕ2(r) , i=1,. . ., 6. (102) And
e(ψ)=
# 1 M+1
(ψ(r)−ψmi,δ(r))2, i=1,. . ., 6. (103)
The relativeL2[0,r0;r2] error is defined by eR(ψ)=
$(ψ(r)−ψmi,δ(r))2
$ψ2(r) , i=1,. . ., 6. (104) Denote Uk=(uk1,uk2,. . .,ukM−1)T,f =(f(r1),f(r2),. . .,f(rM−1))T. Then we obtain the following iterative scheme
AU1=f,
AUk=h(ω1Uk−1+ω2Uk−2+ ã ã ã +ωk−1U1)+f,k=2, 3,. . .,N, (105) whereh= (t)(3−α)1−α,ωi=bi−1−bi, andA=(ai,j)(M−1)×(M−1)is a tridiagonal matrix, here
A(M−1)×(M−1)=
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝
d1 − 1
(r)2a3
2
− 1
(r)2 d2 − 1
(r)2a5
2
. .. . .. . ..
− 1
(r)2 dM−1
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠ ,
where di=hb0+ (r)1 2(2+ 2i), i=1, 2,. . .,M−1, and aj+1
2 =1+ 2j, j=1, 2,. . ., M−2.
Thus we can obtaing(r)=UN by iterative scheme (105).
For the regularized problem, we can also use the finite difference scheme to discretize Equation (48).
In the computational procedure, we takep=1. In discrete format, we takeM=100, N =50 to compute the direct problem. We use the dichotomy method to solve (69), and obtain a posteriori regularization parameter, whereτ =1.1.
Example 5.1: Take functionϕ(r)=sin(r).
In Figures1–3, we give numerical results of Example 5.1 under the a posteriori parame- ter choice rule for various noise levelsε=0.01, 0.008, 0.005 in the case ofα=1.2, 1.5, 1.8.
It can be seen that the numerical error also decreases when the noise is reduced. And the smallerα, the better the approximate effect.
Example 5.2: Take function
ϕ(r)=
⎧⎨
⎩
2r, 0<r< π 2, 2(π−r), π
2 ≤r< π.
In Figures4–6, we give numerical results of Example 5.2 under the a posteriori parameter choice rule for various noise levelsε=0.01, 0.008, 0.005 in the case ofα=1.2, 1.5, 1.8. It can be seen that the numerical error also decreases when the noise is reduced. And the smallerα, the better the approximate effect.
We fixε=0.01. For Table1, whenα=1.2, the iterative steps of Landweber regulariza- tion method, fractional Landweber regularization method and modified iterative method are 597, 75 and 137. Whenα=1.5, the iterative steps of Landweber regularization method, fractional Landweber regularization method and modified iterative method are 272, 41
Figure 1.The exact solution and regular solution of Landweber regularization method by using the a posteriori parameter choice rule for Example 5.1. (a)α=1.2, (b)α=1.5, (c)α=1.8.
Figure 2.The exact solution and regular solution of fractional Landweber regularization method by using the a posteriori parameter choice rule for Example 5.1. (a)α=1.2, (b)α=1.5, (c)α=1.8.
Figure 3.The exact solution and regular solution of modified iterative regularization method by using the a posteriori parameter choice rule for Example 5.1. (a)α=1.2, (b)α=1.5, (c)α=1.8.
Figure 4.The exact solution and regular solution of Landweber regularization method by using the a posteriori parameter choice rule for Example 5.2. (a)α=1.2, (b)α=1.5, (c)α=1.8.
Figure 5.The exact solution and regular solution of fractional Landweber regularization method by using the a posteriori parameter choice rule for Example 5.2. (a)α=1.2, (b)α=1.5, (c)α=1.8.
Figure 6.The exact solution and regular solution of modified iterative regularization method by using the a posteriori parameter choice rule for Example 5.2. (a)α=1.2, (b)α=1.5, (c)α=1.8.
Table 1.The iteration steps of Example 5.1 for different regularization method.
ε 0.01 0.008 0.005
Landweber α=1.2 597 766 1362
α=1.5 272 386 687
α=1.8 262 378 679
Fractional Landweber α=1.2 75 93 138
α=1.5 41 52 85
α=1.8 42 52 83
Modified iterative α=1.2 137 170 261
α=1.5 67 91 155
α=1.8 67 89 153
and 67. Whenα=1.8, the iterative steps of Landweber regularization method, fractional Landweber regularization method and modified iterative method are 262, 42 and 67. We can deduce thatε=0.01 is fixed, the fractional Landweber regularization method has fewer iteration steps. Forε=0.008 andε=0.005, the same result is obtained.
We fixα=1.2. Whenε=0.01, the iterative steps of Landweber regularization method, fractional Landweber regularization method and modified iterative method are 597, 75 and 137. Whenε=0.008, the iterative steps of Landweber regularization method, frac- tional Landweber regularization method and modified iterative method are 766, 93 and 170. Whenε=0.005, the iterative steps of Landweber regularization method, fractional Landweber regularization method and modified iterative method are 1362, 138 and 261.
We can deduce thatα=1.2 is fixed, the fractional
Table 2.The CPU time of Example 5.1 for different regularization method.
α 1.2 1.5 1.8
Landweber ε=0.01 11.9304 5.8485 5.7179
ε=0.008 15.4281 7.4576 7.5216
ε=0.005 27.4445 13.4199 13.4263
Fractional Landweber ε=0.01 1.7742 0.6786 0.7475
ε=0.008 1.5056 1.4258 1.4617
ε=0.005 2.4426 1.4684 1.5987
Modified iterative ε=0.01 2.8694 1.6987 1.7893
ε=0.008 3.4975 1.5797 1.4896
ε=0.005 5.4024 3.4720 3.4510
Landweber regularization method has fewer iteration steps. Forα=1.5 andα=1.8, the same result is obtained.
In summary, the fractional Landweber regularization method has fewer iteration steps.
In Table2, we use a computer with Intel(R) Core(TM) i5-6200U CPU @ 2.30 GHz 2.40 GHz and RAM of 4.00 GB to calculate the CPU time. The specific analysis is as follows:
we fixα=1.2. For Table2, whenε=0.01, the CPU time of Landweber regularization method, fractional Landweber regularization method and modified iterative method are 11.9304 s, 1.7742 s and 2.8694 s. Whenε=0.008, the CPU time of Landweber regulariza- tion method, fractional Landweber regularization method and modified iterative method are 15.4281 s, 1.5056 s and 3.4975 s. Whenε=0.005, the CPU time of Landweber reg- ularization method, fractional Landweber regularization method and modified iterative method are 27.4445 s, 2.4426 s and 5.4024 s. We can deduced thatα=1.2 is fixed, the frac- tional Landweber regularization method has fewer CPU time. Forα=1.5 andα=1.8, the same result is obtained.
We fixε=0.01. Whenα=1.2, the CPU time of Landweber regularization method, fractional Landweber regularization method and modified iterative method are 11.9304 s, 1.7742 s and 2.8694 s. Whenα=1.5, the CPU time of Landweber regularization method, fractional Landweber regularization method and modified iterative method are 5.8485 s, 0.6786 s and 1.6987 s. Whenα=1.8, the CPU time of Landweber regularization method, fractional Landweber regularization method and modified iterative method are 5.7179 s, 0.7475 s and 1.7893 s. We can deduced thatε=0.01 is fixed, the fractional Landweber regularization method has fewer CPU time. Forε=0.008 andε=0.005, the same result is obtained. In summary, the fractional Landweber regularization method has fewer CPU time.
By Table3, we can deduce that the errors between exact solution and approximate solu- tion are smaller for fixedαand the smaller measurement error. And we infer that the error between exact solution and approximation solution of fractional Landweber method is smaller than that obtained by Landweber regularization method and modified iterative method for fixedαandε.
From Table 4, we can obtain the error between exact solution and approximation solution of fractional Landweber method is smaller than that obtained by Landweber regularization method and modified iterative method for fixedαandε.
In Figures1–3, we give numerical results of Example 5.1 under the a posteriori parame- ter choice rule for various noise levelsε=0.01, 0.008, 0.005 in the case ofα=1.2, 1.5, 1.8.
Table 3.Error behaviour of Example 5.1 for differentαwithε=0.01, 0.005.
α 1.1 1.3 1.5 1.7 1.9
Landweber ε=0.01 Rela 0.7320 0.6001 0.5101 0.3923 0.2431
ε=0.01 Abso 0.6078 0.4319 0.3121 0.1920 0.1311
ε=0.005 Rela 0.6501 0.5017 0.3725 0.2519 0.0935
ε=0.005 Abso 0.5910 0.4270 0.2110 0.1175 0.0734
Fractional Landweber ε=0.01 Rela 0.4560 0.3817 0.3039 0.2377 0.2105
ε=0.01 Abso 0.2113 0.1352 0.0987 0.0553 0.0098
ε=0.005 Rela 0.2715 0.1219 0.0902 0.0835 0.0631
ε=0.005 Abso 0.1091 0.0836 0.0430 0.0259 0.0047
Modified iterative ε=0.01 Rela 0.5734 0.5180 0.3945 0.3030 0.2509
ε=0.01 Abso 0.3978 0.2534 0.2575 0.1114 0.0273
ε=0.005 Rela 0.3807 0.3741 0.2370 0.1415 0.0901
ε=0.005 Abso 0.2933 0.1717 0.1682 0.0807 0.0109
Table 4.Error behaviour of Example 5.2 for differentαwithε=0.01, 0.005.
α 1.1 1.3 1.5 1.7 1.9
Landweber ε=0.01 Rela 0.9845 0.7516 0.5322 0.3303 0.2730
ε=0.01 Abso 0.7814 0.5831 0.4911 0.2510 0.1425
ε=0.005 Rela 0.9013 0.6289 0.4516 0.2035 0.1013
ε=0.005 Abso 0.6317 0.5521 0.3849 0.1228 0.0616
Fractional Landweber ε=0.01 Rela 0.4532 0.3907 0.3015 0.2072 0.1723
ε=0.01 Abso 0.3017 0.2125 0.1526 0.0821 0.0101
ε=0.005 Rela 0.2853 0.2036 0.1489 0.0956 0.0745
ε=0.005 Abso 0.1350 0.1049 0.0731 0.0232 0.0059
Modified iterative ε=0.01 Rela 0.6711 0.5332 0.4136 0.2871 0.2020
ε=0.01 Abso 0.3506 0.3029 0.2207 0.1019 0.0310
ε=0.005 Rela 0.3421 0.2521 0.1987 0.1115 0.0109
ε=0.005 Abso 0.1823 0.1352 0.0908 0.0673 0.0456
It can be seen that the numerical error also decreases when the noise is reduced. And the smallerα, the better the approximate effect.
In Figures4–6, we give numerical results of Example 5.2 under the a posteriori parame- ter choice rule for various noise levelsε=0.01, 0.008, 0.005 in the case ofα=1.2, 1.5, 1.8.
It can be seen that the numerical error also decreases when the noise is reduced. And the smallerα, the better the approximate effect.
In Figures7–9, we give numerical results of Example 5.3 under the a posteriori parame- ter choice rule for various noise levelsε=0.01, 0.008, 0.005 in the case ofα=1.2, 1.5, 1.8.
It can be seen that the numerical error also decreases when the noise is reduced. And the smallerα, the better the approximate effect.
Example 5.3: Take function
ϕ(r)=
⎧⎨
⎩
0, 0<r≤ π 2, 1, π
2 <r< π.
In Figures10–12, we give numerical results of Example 5.4 under the a posteriori param- eter choice rule for various noise levelsε=0.01, 0.008, 0.005 in the case ofα=1.2, 1.5, 1.8.
It can be seen that the numerical error also decreases when the noise is reduced. And the smallerα, the better the approximate effect.
Figure 7.The exact solution and regular solution of Landweber regularization method by using the a posteriori parameter choice rule for Example 5.3. (a)α=1.2, (b)α=1.5, (c)α=1.8.
Figure 8.The exact solution and regular solution of fractional Landweber regularization method by using the a posteriori parameter choice rule for Example 5.3. (a)α=1.2, (b)α=1.5, (c)α=1.8.
Figure 9.The exact solution and regular solution of modified iterative regularization method by using the a posteriori parameter choice rule for Example 5.3. (a)α=1.2, (b)α=1.5, (c)α=1.8.
Figure 10.The exact solution and regular solution of Landweber regularization method by using the a posteriori parameter choice rule for Example 5.4. (a)α=1.2, (b)α=1.5, (c)α=1.8.
Figure 11.The exact solution and regular solution of fractional Landweber regularization method by using the a posteriori parameter choice rule for Example 5.4. (a)α=1.2, (b)α=1.5, (c)α=1.8.
Figure 12.The exact solution and regular solution of modified iterative regularization method by using the a posteriori parameter choice rule for Example 5.4. (a)α=1.2, (b)α=1.5, (c)α=1.8.
Figure 13.The exact solution and regular solution of Landweber regularization method by using the a posteriori parameter choice rule for Example 5.5. (a)α=1.2, (b)α=1.5, (c)α=1.8.
In Figures13–15, we give numerical results of Example 5.5 under the a posteriori param- eter choice rule for various noise levelsε=0.01, 0.008, 0.005 in the case ofα=1.2, 1.5, 1.8.
It can be seen that the numerical error also decreases when the noise is reduced. And the smallerα, the better the approximate effect.
In Figures16–18, we give numerical results of Example 5.6 under the a posteriori param- eter choice rule for various noise levelsε=0.01, 0.008, 0.005 in the case ofα=1.2, 1.5, 1.8.
It can be seen that the numerical error also decreases when the noise is reduced. And the smallerα, the better the approximate effect.
Example 5.4: Take functionψ(r)=αcos(r).
Example 5.5: Take function ψ(r)=
⎧⎨
⎩
−2r+π, 0<r≤ π 2, 2r−π, π
2 ≤r< π. Example 5.6: Take function
ψ(r)=
⎧⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎩
1, 0<r≤ π 3, 0, π
3 <r≤ 2π 3 , 1, 2π
3 <r< π.
Figure 14.The exact solution and regular solution of fractional Landweber regularization method by using the a posteriori parameter choice rule for Example 5.5. (a)α=1.2, (b)α=1.5, (c)α=1.8.
Figure 15.The exact solution and regular solution of modified iterative regularization method by using the a posteriori parameter choice rule for Example 5.5. (a)α=1.2, (b)α=1.5, (c)α=1.8.
Figure 16.The exact solution and regularization solution of Landweber regularization method by using the a posteriori parameter choice rule for Example 5.6. (a)α=1.2, (b)α=1.5, (c)α=1.8.
Figure 17.The exact solution and regularization solution of fractional Landweber regularization method by using the a posteriori parameter choice rule for Example 5.6. (a)α=1.2, (b)α=1.5, (c) α=1.8.
Figure 18.The exact solution and regularization solution of modified iterative method by using the a posteriori parameter choice rule for Example 5.6. (a)α=1.2, (b)α=1.5, (c)α=1.8.
Figure 19.The exact solution and regular solution of fractional Landweber regularization method by using the a posteriori parameter choice rule for Example 5.6. (a)α=1.2, (b)α=1.5, (c)α=1.8.
From the above conclusion, we know that the fractional Landweber regularization method is the most efficient and accurate. Next, the error of the regular solution and the exact solution is given in Figure19under the rule of selecting the posterior regularization param- eters when we chooseε=0.01, 0.05, 0.08, respectively. We can know that if the noise levels becomes lager, the error between the exact solution and the regular solution by using the a posteriori parameter choice rule will become lager.
We can deduce that whenεandαare larger, the error between the exact solution and the regular solution by using the a posteriori parameter choice rule is greater.