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An efficient higher-order trigonometric cubic B-spline collocation method for timefractional Burgers equations Cover

An efficient higher-order trigonometric cubic B-spline collocation method for timefractional Burgers equations

International Journal of Mathematics and Computer in Engineering
AHEAD OF PRINT
By: Murat Önal,  Berat Karaagac and  Alaattin Esen  
Open Access
|Mar 2026
Figures & tables

Figures & Tables

Fig. 1

Problem 1: Numerical solutions vs exact solutions and γ values.

Fig. 2

Problem 1: (a) Evolution of numerical solutions vs time levels, (b) 3D graphs of numerical solutions.

Fig. 3

Problem 1: The absolute errors for N = 100, γ = 0.5, Δt = 0.001 at tf = 1.

Fig. 4

Problem 2: (a) Numerical solutions vs exact solutions, (b) Numerical solutions vs γ values.

Fig. 5

Problem 2: (a) Evolution of numerical solutions vs time levels, (b) 3D graphs of numerical solutions.

Fig. 6

Problem 2: The absolute errors for N = 100, γ = 0.5,∆t = 0.001 at tf = 1.

Fig. 7

Problem 3: (a) Numerical solutions vs exact solutions, (b) Numerical solutions vs γ values.

Fig. 8

Problem 3: (a) Evolution of numerical solutions vs time levels, (b) 3D graphs of numerical solutions.

Fig. 9

Problem 3: The absolute Errors for N = 100, γ = 0.5,∆t = 0.001 at tf = 1.

Comparison of error norms for γ = 0_5, N = 120, tf = 1 and different values of Δt for Problem 1_

∆tNormProposed methodRef. [10]Ref. [11]
0.002L2L∞2.89 × 10−54.09 × 10−51.22 × 10−31.72 × 10−35.55 × 10−58.01 × 10−5
0.001L2L∞9.70 × 10−61.37 × 10−55.32 × 10−47.53 × 10−42.77 × 10−53.92 × 10−5
0.0005L2L∞1.18 × 10−82.09 × 10−81.89 × 10−42.68 × 10−41.39 × 10−52.05 × 10−5

The error norms L2 and L∞ for varying values of γ and N for ∆t = 0_0005 of Problem 2_

NNormγ = 0.25γ = 0.5γ = 0.75γ = 0.9
20L2L∞1.61504086 × 10−43.08801388 × 10−41.44824109 × 10−41.98520497 × 10−41.38338571 × 10−41.89620989 × 10−41.15834339 × 10−41.58724458 × 10−4
40L2L∞8.55740423 × 10−51.17333257 × 10−48.82849416 × 10−51.21056287 × 10−48.27624785 × 10−51.13520309 × 10−46.08117897 × 10−58.34162090 × 10−5
80L2L∞7.11925979 × 10−59.76772696 × 10−57.41488030 × 10−51.01737613 × 10−46.88664654 × 10−59.45207070 × 10−54.70544770 × 10−56.45980625 × 10−5
100L2L∞6.94667643 × 10−59.53080411 × 10−57.24524065 × 10−59.94124999 × 10−56.71988959 × 10−59.22404235 × 10−54.54035631 × 10−56.23420557 × 10−5

The error norms L2 and L∞ for varying values of γ and Δt for N = 120 of Problem 3_

∆tNormγ = 0.25γ = 0.5γ = 0.75γ = 0.9
0.002L2L∞3.14766794 × 10−54.39151250 × 10−53.22741925 × 10−54.50291619 × 10−52.77834353 × 10−53.87661038 × 10−51.62793905 × 10−52.27160596 × 10−5
0.001L2L∞1.46307138 × 10−52.04121181 × 10−51.51160676 × 10−52.10899969 × 10−51.31564294 × 10−51.83571994 × 10−57.53330880 × 10−61.05120674 × 10−5
0.0005L2L∞6.23879040 × 10−68.70409358 × 10−66.51771718 × 10−69.09364603 × 10−65.66868928 × 10−67.90976865 × 10−62.93671313 × 10−64.09826535 × 10−6

The error norms L2 and L∞ for varying values of γ and Δt for N = 120 of Problem 2_

∆tNormγ = 0.25γ = 0.5γ = 0.75γ = 0.9
0.002L2L∞2.66079805 × 10−43.65024346 × 10−42.75969290 × 10−43.78638010 × 10−42.46238628 × 10−43.37996968 × 10−41.54248522 × 10−42.11803042 × 10−4
0.001L2L∞1.34644789 × 10−41.84721757 × 10−41.40213575 × 10−41.92384206 × 10−41.27686564 × 10−41.75274357 × 10−48.28012104 × 10−51.13702401 × 10−4
0.0005L2L∞6.85292690 × 10−59.40182449 × 10−57.15309025 × 10−59.81468420 × 10−56.62930520 × 10−59.09997581 × 10−54.45067678 × 10−56.11154536 × 10−5

Comparison of errors for γ = 0_5, N = 120, tf = 1 and different values of Δt (Problem 3)_

∆tMethodL2L∞
0.002Proposed methodRef. [10]Ref. [11]3.22 × 10−51.22 × 10−35.55 × 10−54.50 × 10−51.72 × 10−38.01 × 10−5
0.001Proposed methodRef. [10]Ref. [11]1.51 × 10−55.32 × 10−42.77 × 10−52.10 × 10−57.53 × 10−43.92 × 10−5
0.0005Proposed methodRef. [10]Ref. [11]6.51 × 10−61.89 × 10−41.39 × 10−59.09 × 10−62.68 × 10−42.05 × 10−5

Comparison of errors for γ = 0_5, ∆t = 0_00025, tf = 1 and different values of N (Problem 3)_

N MethodMethodL2L∞
40Proposed methodRef. [10]Ref. [11]1.47 × 10−51.22 × 10−31.60 × 10−52.05 × 10−51.73 × 10−32.63 × 10−5
80Proposed methodRef. [10]Ref. [11]4.41 × 10−71.78 × 10−47.72 × 10−66.15 × 10−72.53 × 10−41.34 × 10−5
100Proposed methodRef. [10]Ref. [11]1.27 × 10−65.23 × 10−57.24 × 10−61.77 × 10−67.65 × 10−51.19 × 10−5

The error norms for Δt = 0_0005 and different values of N and γ of Problem 1_

NNormγ = 0.25γ = 0.5γ = 0.75γ = 0.9
10L2L∞1.17608481 × 10−31.56691086 × 10−31.16808997 × 10−31.55586455 × 10−31.16122058 × 10−31.54633448 × 10−31.15977847 × 10−31.54428531 × 10−3
20L2L∞3.24966594 × 10−44.58600285 × 10−43.22510891 × 10−44.55139112 × 10−43.21299945 × 10−44.53437127 × 10−43.23144788 × 10−44.56047172 × 10−4
40L2L∞7.78030654 × 10−51.09833692 × 10−47.69389503 × 10−51.08615283 × 10−47.73445392 × 10−51.09189833 × 10−48.01209788 × 10−51.13110804 × 10−4
80L2L∞1.25916352 × 10−51.78288364 × 10−51.21469751 × 10−51.71989088 × 10−51.29784520 × 10−51.83758881 × 10−51.60002218 × 10−52.26544495 × 10−5

Maximum errors and convergence rates for γ = 0_5, v = 1, tf = 1 for different values of N and Δt_

N∆tL∞ RoC
101/45.26044936 × 10−3−
201/323.50923812 × 10−43.90
401/2561.65112329 × 10−54.40

The error norms L2 and L∞ for varying values of γ and N for Δt = 0_00025 of Problem 3_

NNormγ = 0.1γ = 0.2γ = 0.4γ = 0.6
10L2L∞3.01229397 × 10−44.14249903 × 10−43.00321184 × 10−44.13024518 × 10−42.98502812 × 10−44.10575706 × 10−42.96792891 × 10−44.08282315 × 10−4
20L2L∞7.28223128 × 10−51.00695720 × 10−47.24982978 × 10−51.00239858 × 10−47.19129216 × 10−59.94134006 × 10−57.14959653 × 10−59.88181649 × 10−5
40L2L∞1.52632425 × 10−52.12934859 × 10−51.50860659 × 10−52.10460160 × 10−51.48106479 × 10−52.06611232 × 10−51.47187945 × 10−52.05322390 × 10−5
80L2L∞8.18623973 × 10−71.14201244 × 10−66.78279439 × 10−79.46133375 × 10−74.80627584 × 10−76.70126490 × 10−74.70380184 × 10−76.55416123 × 10−7

The error norms for N = 120 and different values of Δt and γ at T = 1 for Problem 1_

∆tNormγ = 0.25γ = 0.5γ = 0.75γ = 0.9
0.002L2L∞2.79799013 × 10−53.96292257 × 10−52.89453357 × 10−54.09955552 × 10−52.39205481 × 10−53.38827254 × 10−51.09172130 × 10−51.54759469 × 10−5
0.001L2L∞9.09662936 × 10−61.28824323 × 10−59.70022354 × 10−61.37368370 × 10−57.53455829 × 10−61.06714152 × 10−51.19234428 × 10−61.69358924 × 10−6
0.0005L2L∞3.57839140 × 10−75.05989927 × 10−71.18088684 × 10−82.09713397 × 10−89.03171600 × 10−71.27789156 × 10−63.97091243 × 10−65.62041476 × 10−6

Comparison of numerical solutions and errors for γ = 0_5, ∆t = 0_00025, tf = 1, ν = 1 and N = 40 for Problem 2_

xProposed methodL∞Ref. [12]L∞Exact solution
0.21.2213663.66 × 10−51.2214625.95 × 10−51.221402
0.41.4917626.24 × 10−51.4919341.09 × 10−41.491824
0.61.8220457.36 × 10−51.8222581.39 × 10−41.822118
0.82.2254806.05 × 10−52.2256661.025 × 10−42.225540

Comparison of errors for γ = 0_5, ∆t = 0_00025, tf = 1 and different values of N for Problem 2_

NMethodL2L∞
40Proposed methodRef. [10]5.36 × 10−56.77 × 10−57.36 × 10−52.09 × 10−4
80Proposed methodRef. [10]3.95 × 10−54.57 × 10−55.42 × 10−56.92 × 10−5

The error norms L∞ and convergence rates for varying values of N and Δt for γ = 0_9, ν = 1, tf = 1 for Problem 2_

N∆tL∞RoC
1001/42.20820268 × 10−2−
2001/641.20495216 × 10−34.19
4001/10241.08765049 × 10−43.46

Maximum errors and convergence rates for γ = 0_5, ν = 1, tf = 1 and different values of Δt and N (Problem 3)_

N∆tL∞RoC
121/51.64465660 × 10−4−
241/407.17815902 × 10−63.64
481/3207.21429488 × 10−73.25
961/25603.72153127 × 10−83.55

Comparison of error norms for γ = 0_5, Δt = 0_00025, tf = 1 and different values of N for Problem 1_

NNormProposed methodRef. [10]Ref. [11]
40L2L∞8.18 × 10−51.15 × 10−41.22 × 10−31.73 × 10−31.60 × 10−52.63 × 10−5
80L2L∞1.70 × 10−52.40 × 10−51.78 × 10−42.53 × 10−47.72 × 10−61.34 × 10−5
100L2L∞9.14 × 10−61.29 × 10−55.23 × 10−57.65 × 10−57.24 × 10−61.19 × 10−5
DOI: https://doi.org/10.2478/ijmce-2026-0013 | Journal eISSN: 2956-7068
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Language: English
Submitted on: Jan 7, 2026
Accepted on: Feb 12, 2026
Published on: Mar 18, 2026
Published by: Harran University
In partnership with: Paradigm Publishing Services
Publication frequency: 2 issues per year
Keywords:
Time-fractional Burgers equation,
higher-order method,
trigonometric cubic B-spline,
collocation method,
stability
Related subjects:
Computer sciences,
Computer sciences, other,
Engineering,
Introductions and overviews,
Engineering, other,
Mathematics,
General mathematics,
Physics,
Physics, other

© 2026 Murat Önal, Berat Karaagac, Alaattin Esen, published by Harran University
This work is licensed under the Creative Commons Attribution 4.0 License.

AHEAD OF PRINT