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A study on different classes of differential equations by semi-analytical and numerical techniques Cover

A study on different classes of differential equations by semi-analytical and numerical techniques

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

Comparison of the exact solution with HAM and HWT solutions for Application 4.1.

Fig. 2

Error analysis of HAM and HWT solutions for Application 4.1.

Fig. 3

Comparison of the ND solver solution with HAM and HWT solutions for Application 4.2.

Fig. 4

Error analysis of HAM and HWT solutions for Application 4.2.

Fig. 5

Comparison of the exact solution with HAM and HWT solutions for u1(x) in Application 4.3.

Fig. 6

Error analysis of HAM and HWT solutions for u1(x) in Application 4.3.

Fig. 7

Comparison of the exact solution with HAM and HWT solutions for u2 (x) in Application 4.3.

Fig. 8

Error analysis of HAM and HWT solutions for u2 (x) in Application 4.3.

Fig. 9

Comparison of the exact solution with HAM and HWT solutions at x = 0 of Application 4.4.

Fig. 10

Error analysis of HAM and HWT solutions at x = 0 of Application 4.4.

Fig. 11

Comparison of the exact solution with HAM and HWT solutions at x = 1 of Application 4.4.

Fig. 12

Error analysis of HAM and HWT solutions at x = 1 of Application 4.4.

Fig. 13

Comparison of the exact solution with HAM and HWT solutions at t = 0 of Application 4.4.

Fig. 14

Error analysis of HAM and HWT solutions at t = 0 of Application 4.4.

Fig. 15

Comparison of the exact solution with HAM and HWT solutions at t = 1 of Application 4.4.

Fig. 16

Error analysis of HAM and HWT solutions at t = 1 of Application 4.4.

Fig. 17

3D plot of HAM and HWT solutions for various values of x and t of Application 4.4.

Fig. 1

Comparison of the exact solution with HAM and HWT solutions for Application 4.1.

Fig. 2

Error analysis of HAM and HWT solutions for Application 4.1.

Fig. 3

Comparison of the ND solver solution with HAM and HWT solutions for Application 4.2.

Fig. 4

Error analysis of HAM and HWT solutions for Application 4.2.

Fig. 5

Comparison of the exact solution with HAM and HWT solutions for u1(x) in Application 4.3.

Fig. 6

Error analysis of HAM and HWT solutions for u1(x) in Application 4.3.

Fig. 7

Comparison of the exact solution with HAM and HWT solutions for u2(x) in Application 4.3.

Fig. 8

Error analysis of HAM and HWT solutions for u2(x) in Application 4.3.

Fig. 9

Comparison of the exact solution with HAM and HWT solutions at x = 0 of Application 4.4.

Fig. 10

Error analysis of HAM and HWT solutions at x = 0 of Application 4.4.

Fig. 11

Comparison of the exact solution with HAM and HWT solutions at x = 1 of Application 4.4.

Fig. 12

Error analysis of HAM and HWT solutions at x = 1 of Application 4.4.

Fig. 13

Comparison of the exact solution with HAM and HWT solutions at t = 0 of Application 4.4.

Fig. 14

Error analysis of HAM and HWT solutions at t = 0 of Application 4.4.

Fig. 15

Comparison of the exact solution with HAM and HWT solutions at t = 1 of Application 4.4.

Fig. 16

Error analysis of HAM and HWT solutions at t = 1 of Application 4.4.

Fig. 17

3D plot of HAM and HWT solutions for various values of x and t of Application 4.4.

Comparison of absolute error AE of the HAM and HWT solutions with Exact solution for x = 0_1, x = 0_01, and x = 0_001 in Application 4_4_

tError at x = 0.1Error at x = 0.01Error at x = 0.001
HAMHWTHAMHWTHAMHWT
0000000
0.105.671 × 10-605.672 × 10-905.671 × 10-12
0.202.268 × 10-502.268 × 10-802.268 × 10-11
0.305.104 × 10-505.104 × 10-805.104 × 10-11
0.409.075 × 10-509.075 × 10-809.075 × 10-11
0.501.418 × 10-401.418 × 10-701.418 × 10-10
0.602.041 × 10-402.041 × 10-702.041 × 10-10
0.702.779 × 10-402.779 × 10-702.779 × 10-10
0.803.630 × 10-403.630 × 10-703.630 × 10-10
0.904.594 × 10-404.594 × 10-704.594 × 10-10
1.005.672 × 10-405.672 × 10-705.672 × 10-10

Comparison of solutions obtained from HAM, HWT, and their AE with ND Solver solution for Application 4_2_

xND SolverHAMHWTHAM ErrorHWT Error
00.50.50.50.0
0.10.524380.524380.524392.048 × 10-81.120 × 10-6
0.20.547580.547580.547591.718 × 10-84.480 × 10-6
0.30.569640.569640.569661.413 × 10-81.008 × 10-5
0.40.590630.590630.590659.894 × 10-91.792 × 10-5
0.50.610590.610590.610632.164 × 10-82.800 × 10-5
0.60.629590.629590.629631.587 × 10-84.032 × 10-5
0.70.647650.647650.647714.028 × 10-85.488 × 10-5
0.80.664830.664830.664911.566 × 10-87.168 × 10-5
0.90.681180.681180.681281.790 × 10-89.072 × 10-5
1.00.696730.696730.696857.461 × 10-101.120 × 10-4

Comparison of solutions obtained from HAM, HWT, and their AE with Exact solution for u1 in Application 4_3_

xExactHAMHWTHAM ErrorHWT Error
011100
0.10.670320.670320.6703201.120 × 10-7
0.20.449320.449320.4493308.960 × 10-7
0.30.301190.301190.3012003.024 × 10-6
0.40.201890.201890.2019007.168 × 10-6
0.50.135330.135330.1353501.140 × 10-5
0.60.090710.090710.0907402.419 × 10-5
0.70.060810.060810.0608403.841 × 10-5
0.80.040760.040760.0408205.734 × 10-5
0.90.027320.027320.0274008.164 × 10-5
1.00.018310.018310.0184201.120 × 10-4

Comparison of solutions obtained from HAM, HWT, and their AE with Exact solution for x = 0 in Application 4_4_

tExactHAMHWTHAM ErrorHWT Error
011100
0.11.105171.105171.1051700
0.21.221401.221401.2214000
0.31.349861.349861.3498600
0.41.491821.491821.4918200
0.51.648721.648721.6487200
0.61.822121.822121.8221200
0.72.013752.013752.0137500
0.82.225542.225542.2255400
0.92.459602.459602.4596000
1.02.718282.718282.7182800

Comparison of solutions obtained from HAM, HWT, and their AE with Exact solution for x = 1 in Application 4_4_

tExactHAMHWT ErrorHAM ErrorHWT
00.367870.367870.3678700
0.10.406570.406570.4122405.672 × 10-3
0.20.449320.449320.4720202.268 × 10-2
0.30.496580.496580.5476305.104 × 10-2
0.40.548810.548810.6395609.075 × 10-2
0.50.606530.606530.7483301.418 × 10-1
0.60.670320.670320.8745102.042 × 10-1
0.70.740810.740811.0187002.779 × 10-1
0.80.818730.818731.1817503.630 × 10-1
0.90.904830.904831.3643204.594 × 10-1
1.0111.5672105.672 × 10-1

Comparison of solutions obtained from HAM, HWT, and their AE with Exact solution for t = 0 in Application 4_4_

xExactHAMHWT ErrorHAM ErrorHWT
011100
0.10.904830.904830.9048300
0.20.818730.818730.8187300
0.30.740810.740810.7408100
0.40.670320.670320.6703200
0.50.606530.606530.6065300
0.60.548810.548810.5488100
0.70.496580.496580.4965800
0.80.449320.449320.4493200
0.90.406560.406560.4065600
1.00.367870.367870.3678700

Comparison of solutions obtained from HAM, HWT, and their AE with Exact solution for t = 1 in Application 4_4_

xExactHAMHWT ErrorHAM ErrorHWT
02.718282.718282.7182800
0.12.459602.459602.4602105.672 × 10-4
0.22.225542.225542.2301204.357 × 10-3
0.32.013752.013752.029101.531 × 10-2
0.41.822121.822121.858403.630 × 10-2
0.51.648721.648721.719607.090 × 10-2
0.61.491821.491821.614301.225 × 10-1
0.71.349861.349861.544401.945 × 10-1
0.81.221401.221401.511802.904 × 10-1
0.91.105171.105171.518704.134 × 10-1
1.0111.567205.672 × 10-1

Comparison of solutions obtained from HAM, HWT, and their absolute errors (AE) with Exact solution for Application 4_1_

xExactHAMHWTHAM ErrorHWT Error
0.50.5410490.5409740.541477.554 × 10-55.0 × 10-4
0.60.5816220.5816180.582344.062 × 10-67.2 × 10-4
0.70.6143610.6143610.615343.496 × 10-79.8 × 10-4
0.80.6399260.6399260.641212.917 × 10-71.28 × 10-3
0.90.6588130.6588130.660432.933 × 10-71.62 × 10-3
1.00.6713970.6713970.673402.929 × 10-72 × 10-3
1.10.6779890.6779890.680412.904 × 10-72.42 × 10-3
1.20.6788660.6788650.681752.838 × 10-72.88 × 10-3
1.30.6742860.6742890.677673.718 × 10-63.38 × 10-3
1.40.6635340.6645240.668449.9 × 10-43.92 × 10-3

Comparison of absolute error AE of the HAM and HWT solutions with Exact solution for t = 0_1, t = 0_01, and t = 0_001 in Application 4_4_

xError at t = 0.1Error at t = 0.01Error at t = 0.001
HAMHWTHAMHWTHAMHWT
0000000
0.105.672 × 10-605.672 × 10-805.672 × 10-10
0.204.537 × 10-504.537 × 10-704.537 × 10-9
0.301.531 × 10-401.531 × 10-601.531 × 10-8
0.403.630 × 10-403.630 × 10-603.630 × 10-8
0.507.090 × 10-407.090 × 10-607.090 × 10-8
0.601.225 × 10-301.225 × 10-501.225 × 10-7
0.701.945 × 10-301.945 × 10-501.945 × 10-7
0.802.904 × 10-302.904 × 10-502.904 × 10-7
0.904.134 × 10-304.134 × 10-504.134 × 10-7
1.005.672 × 10-305.672 × 10-505.672 × 10-7

Comparison of solutions obtained from HAM, HWT, and their AE with Exact solution for u2 in Application 4_3_

xND SolverHAMHWTHAM ErrorHWT Error
00.50.50.50.0
0.10.524380.524380.524392.048 × 10-81.120 × 10-6
0.20.547580.547580.547591.718 × 10-84.480 × 10-6
0.30.569640.569640.569661.413 × 10-81.008 × 10-5
0.40.590630.590630.590659.894 × 10-91.792 × 10-5
0.50.610590.610590.610632.164 × 10-82.800 × 10-5
0.60.629590.629590.629631.587 × 10-84.032 × 10-5
0.70.647650.647650.647714.028 × 10-85.488 × 10-5
0.80.664830.664830.664911.566 × 10-87.168 × 10-5
0.90.681180.681180.681281.790 × 10-89.072 × 10-5
1.00.696730.696730.696857.461 × 10-101.120 × 10-4
DOI: https://doi.org/10.2478/ijmce-2026-0004 | Journal eISSN: 2956-7068
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Language: English
Page range: 45 - 64
Submitted on: Sep 16, 2024
Accepted on: Jan 12, 2025
Published on: May 27, 2026
Published by: Harran University
In partnership with: Paradigm Publishing Services
Publication frequency: 2 issues per year
Keywords:
Ordinary differential equations,
HAM,
Haar wavelet,
PDES,
HWT
Related subjects:
Computer sciences,
Computer sciences, other,
Engineering,
Introductions and overviews,
Engineering, other,
Mathematics,
General mathematics,
Physics,
Physics, other

© 2026 Sachin K. Narayana, Suguntha Devi Kannadasan, Kumbinarasaiah Srinivasa, 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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