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

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

International Journal of Mathematics and Computer in Engineering
Volume 4 (2026): Issue 1 (June 2026)
By: ,   and    
Open Access
|May 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.

The error norms Le and convergence rates for varying values of N and Ar for γ=0_9, v=1, tf=1 for Problem 2_

NAtL∞RoC
1001/42.20820268×10^{-2}-
2001/641.20495216×10-34.19
4001/10241.08765049×10-43.46

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

Δ tMethodL2L∞
0.002Proposed method3.22×10-54.50×10-5
Ref. [10]1.22×10-31.72×10-3
Ref. [11]5.55×10-58.01×10-5
0.001Proposed method1.51×10-52.10×10-5
Ref. [10]5.32×10-47.53×10-4
Ref. [11]2.77×10-53.92×10-5
0.0005Proposed method6.51×10-69.09×10-6
Ref. [10]1.89×10-42.68×10-4
Ref. [11]1.39×10-52.05×10-5

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

Δ tNormγ=0.25γ=0.5γ=0.75γ=0.9
0.002L22.79799013×10-52.89453357×10-52.39205481×10-51.09172130×10-5
L∞3.96292257×10-54.09955552×10-53.38827254×10-51.54759469×10-5
0.001L29.09662936×10-61.28824323×10-57.53455829×10-61.19234428×10-6
L∞9.70022354×10-61.37368370×10-51.06714152×10-51.69358924×10-6
0.0005L21.18088684×10-83.57839140×10-79.03171600×10-73.97091243×10-6
L∞2.09713397×10-85.05989927×10-71.27789156×10-65.62041476×10-6

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

NMethodsL2L∞
40Proposed method5.36×10-57.36×10-5
Ref. [10]6.77×10-52.09×10-4
80Proposed method3.95×10-55.42×10-5
Ref. [10]4.57×10-56.92×10-5

The error norms L₂ and Los for varying values of y and Ar for N=120 of Problem 3_

Δ tNormγ=0.25γ=0.5γ=0.75γ=0.9
0.002L23.14766794×10-53.22741925×10-52.77834353×10-51.62793905×10-5
L∞4.39151250×10-54.50291619×10-53.87661038×10-52.27160596×10-5
0.001L21.46307138×10-51.51160676×10-51.31564294×10-57.5330880×10-6
L∞2.04121181×10-52.10899969×10-51.83571994×10-51.05120674×10-5
0.0005L26.23879040×10-66.51771718×10-65.66868928×10-62.93671313×10-6
L∞8.70409358×10-69.09364603×10-67.90976865×10-64.09826535×10-6

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

NΔ tL∞ROC
101/45.26044936×10-3-
201/323.50923812×10-43.90
401/2561.65112329×10-54.40

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

Δ tNormProposed methodRef. [10]Ref. [11]
0.002L22.89×10-51.22×10-55.55×10-5
L∞4.09×10-51.72×10-38.01×10-5
0.001L29.70×10-65.32×10-42.77×10-5
L∞1.37×10-57.53×10-43.92×10-5
0.0005L21.18×10-81.89×10-41.39×10-5
L∞2.09×10-82.68×10-42.05×10-5

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

NΔ tL∞ROC
81/56.43241077×10-3-
161/404.49763147×10-43.83
321/3203.33582433×10-53.75

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

NMethodL2Loo
40Proposed method1.47×10-52.05×10-5
Ref. [10]1.22×10-31.73×10-3
Ref. [11]1.60×10-52.63×10-5
80Proposed method4.41×10-76.15×10-7
Ref. [10]1.78×10-42.53×10-4
Ref. [11]7.72×10-61.34×10-5
100Proposed method1.27×10-61.77×10-6
Ref. [10]5.23×10-57.65×10-5
Ref. [11]7.24×10-61.19×10-5

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

NNormProposed methodRef. [10]Ref. [11]
40L28.18×10-51.22×10-31.60×10-5
L∞1.15×10-41.73×10-32.63×10-5
80L21.70×10-51.78×10-47.72×10-6
L∞2.40×10-52.53×10-41.34×10-5
100L29.14×10-65.23×10-57.24×10-6
L∞1.29×10-57.65×10-51.19×10-5

Maximum errors and convergence rates for y = 0_5, v=1, tf=1 and different values of Ar and N (Problem 3)_

NΔ 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

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

NNormγ=0.25γ=0.5γ=0.75γ=0.9
10L21.17608481×10-31.16808997×10-31.16122058×10-31.15977847×10-3
L∞1.56691086×10-31.55586455×10-31.54633448×10-31.54428531×10-3
20L23.24966594×10-43.22510891×10-43.21299945×10-43.23144788×10-4
L∞4.58600285×10-44.55139112×10-44.53437127×10-44.56047172×10-4
40L27.78030654×10-57.69389503×10-57.73445392×10-58.01209788×10-5
L∞1.09833692×10-41.08615283×10-41.09189833×10-41.13110804×10-4
80L21.25916352×10-51.21469751×10-51.29784520×10-51.60002218×10-5
L∞1.78288364×10-51.71989088×10-51.83758881×10-52.26544495×10-5

The error norms L₂ and L∞_ for varying values of γ and 394;t for N = 120 of Problem 2_

Δ tNormγ=0.25γ=0.5γ=0.75γ=0.9
0.002L22.66079805×10-42.75969290×10-42.46238628×10-41.54248522×10-4
L∞3.65024346×10-43.78638010×10-43.37996968×10-42.11803042×10-4
0.001L21.34644789×10-41.40213575×10-41.27686564×10-48.28012104×10-5
L∞1.84721757×10-41.92384206×10-41.75274357×10-41.13702401×10-4
0.0005L26.85292690×10-57.15309025×10-56.62930520×10-54.45067678×10-5
L∞9.40182449×10-59.81468420×10-59.09997581×10-56.11154536×10-5

The error norms L₂ and L_ for varying values of y and N for Δ t=0_00025 of Problem 3_

NNormγ=0.1γ=0.2γ=0.4γ=0.6
10L23.01229397×10-43.00321184×10-42.98502812×10-42.96792891×10-4
L∞4.14249903×10-44.13024518×10-44.10575706×10-44.08282315×10-4
20L27.28223128×10-57.24982978×10-57.19129216×10-57.14959653×10-5
L∞1.00695720×10-41.00239858×10-49.94134006×10-59.88181649×10-5
40L21.52632425×10-51.50860659×10-51.48106479×10-51.47187945×10-5
L∞2.12934859×10-52.10460160×10-52.06611232×10-52.05322390×10-5
80L28.18623973×10-76.78279439×10-74.80627584×10-74.70380184×10-7
L∞1.14201244×10-69.46133375×10-76.70126490×10-76.55416123×10-7

Maximum errors and convergence rates for y = 0_5, v = 1, tf = 1 and different values of Ar and N (Problem 3)_

NΔ tL∞ROC
51/47.90193068×10-3-
101/323.54754064×10-44.47
201/2561.22189859×10-54.85

The error norms L_ and convergence rates for varying values of N and Ar for γ=0_9, v=1, tf=1 for Problem 2_

NΔ tL∞ROC
301/42.20295625×10^{-2}-
601/641.21511503×10-34.18
1201/10241.11304212×10-43.44

Comparison of numerical solutions and errors for γ=0_5, Δt=0_00025, t = 1, v=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

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

NNormγ=0.25γ=0.5γ=0.75γ=0.9
20L21.61504086×10-41.44824109×10-41.38338571×10-41.15834339×10-4
L∞3.08801388×10-41.98520497×10-41.89620989×10-41.58724458×10-4
40L28.55740423×10-58.82849416×10-58.27624785×10-56.08117897×10-5
L∞1.17333257×10-41.21056287×10-41.13520309×10-48.34162090×10-5
80L27.11925979×10-57.41488030×10-56.88664654×10-54.70544770×10-5
L∞9.76772696×10-51.01737613×10-49.45207070×10-56.45980625×10-5
100L26.94667643×10-57.24524065×10-56.71988959×10-54.54035631×10-5
L∞9.53080411×10-59.94124999×10-59.22404235×10-56.23420557×10-5

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

NΔ tL∞ROC
61/54.51974778×10-3-
121/201.64465660×10-44.78
241/807.17815902×10-64.51
481/3207.21429488×10-73.31
961/12803.72153127×10-84.27
DOI: https://doi.org/10.2478/ijmce-2026-0013 | Journal eISSN: 2956-7068
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Language: English
Page range: 211 - 234
Submitted on: Jan 7, 2026
Accepted on: Feb 12, 2026
Published on: May 27, 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.

Volume 4 (2026): Issue 1 (June 2026)
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