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

Narayana, Sachin K.; Devi Kannadasan, Suguntha; Srinivasa, Kumbinarasaiah

doi:10.2478/ijmce-2026-0004

Figures & Tables

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

Error analysis of HAM and HWT solutions for Application 4.1.

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

Error analysis of HAM and HWT solutions for Application 4.2.

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

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

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

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

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

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

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

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

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

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

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

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

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_

t	Error at x = 0.1		Error at x = 0.01		Error at x = 0.001
	HAM	HWT	HAM	HWT	HAM	HWT
0	0	0	0	0	0	0
0.1	0	5.671 ×10⁻⁶	0	5.672×10⁻⁹	0	5.671 ×10⁻¹²
0.2	0	2.268 ×10⁻⁵	0	2.268 ×10⁻⁸	0	2.268 ×10⁻¹¹
0.3	0	5.104×10⁻⁵	0	5.104×10⁻⁸	0	5.104×10⁻¹¹
0.4	0	9.075×10⁻⁵	0	9.075×10⁻⁸	0	9.075×10⁻¹¹
0.5	0	1.418×10⁻⁴	0	1.418×10⁻⁷	0	1.418×10⁻¹⁰
0.6	0	2.041 ×10⁻⁴	0	2.041 ×10⁻⁷	0	2.041 ×10⁻¹⁰
0.7	0	2.779 ×10⁻⁴	0	2.779 ×10⁻⁷	0	2.779 ×10⁻¹⁰
0.8	0	3.630×10⁻⁴	0	3.630×10⁻⁷	0	3.630×10⁻¹⁰
0.9	0	4.594× 10⁻⁴	0	4.594×10⁻⁷	0	4.594×10⁻¹⁰
1.0	0	5.672× 10⁻⁴	0	5.672×10⁻⁷	0	5.672×10⁻¹⁰

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

x	ND Solver	HAM	HWT	HAM Error	HWT Error
0	0.5	0.5	0.5	0.	0
0.1	0.52438	0.52438	0.52439	2.048 ×10⁻⁸	1.120 ×10⁻⁶
0.2	0.54758	0.54758	0.54759	1.718×10⁻⁸	4.480 ×10⁻⁶
0.3	0.56964	0.56964	0.56966	1.413×10⁻⁸	1.008 ×10⁻⁵
0.4	0.59063	0.59063	0.59065	9.894×10⁻⁹	1.792 ×10⁻⁵
0.5	0.61059	0.61059	0.61063	2.164×10⁻⁸	2.800×10⁻⁵
0.6	0.62959	0.62959	0.62963	1.587×10⁻⁸	4.032×10⁻⁵
0.7	0.64765	0.64765	0.64771	4.028 ×10⁻⁸	5.488 ×10⁻⁵
0.8	0.66483	0.66483	0.66491	1.566 ×10⁻⁸	7.168 ×10⁻⁵
0.9	0.68118	0.68118	0.68128	1.790 ×10⁻⁸	9.072 ×10⁻⁵
1.0	0.69673	0.69673	0.69685	7.461 ×10⁻¹⁰	1.120 ×10⁻⁴

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

t	Exact	HAM	HWT
0	1	1	1
0.1	1.10517	1.10517	1.10517
0.2	1.22140	1.22140	1.22140
0.3	1.34986	1.34986	1.34986
0.4	1.49182	1.49182	1.49182
0.5	1.64872	1.64872	1.64872
0.6	1.82212	1.82212	1.82212
0.7	2.01375	2.01375	2.01375
0.8	2.22554	2.22554	2.22554
0.9	2.45960	2.45960	2.45960
1.0	2.71828	2.71828	2.71828

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

x	Exact	HAM	HWT	HWT Error
0	1	1	1	0
0.1	0.67032	0.67032	0.67032	1.120 ×10⁻⁷
0.2	0.44932	0.44932	0.44933	8.960×10⁻⁷
0.3	0.30119	0.30119	0.30120	3.024×10⁻⁶
0.4	0.20189	0.20189	0.20190	7.168 ×10⁻⁶
0.5	0.13533	0.13533	0.13535	1.140 ×10⁻⁵
0.6	0.09071	0.09071	0.09074	2.419 ×10⁻⁵
0.7	0.06081	0.06081	0.06084	3.841 ×10⁻⁵
0.8	0.04076	0.04076	0.04082	5.734×10⁻⁵
0.9	0.02732	0.02732	0.02740	8.164×10⁻⁵
1.0	0.01831	0.01831	0.01842	1.120 ×10⁻⁴

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_

x	Error at t = 0.1		Error at t = 0.01		Error at t = 0.001
	HAM	HWT	HAM	HWT	HAM	HWT
0	0	0	0	0	0	0
0.1	0	5.672×10⁻⁶	0	5.672×10⁻⁸	0	5.672×10⁻¹⁰
0.2	0	4.537×10⁻⁵	0	4.537×10⁻⁷	0	4.537×10⁻⁹
0.3	0	1.531×10⁻⁴	0	1.531×10⁻⁶	0	1.531×10⁻⁸
0.4	0	3.630×10⁻⁴	0	3.630×10⁻⁶	0	3.630×10⁻⁸
0.5	0	7.090× 10⁻⁴	0	7.090× 10⁻⁶	0	7.090× 10⁻⁸
0.6	0	1.225 ×10⁻³	0	1.225 ×10⁻⁵	0	1.225 ×10⁻⁷
0.7	0	1.945 ×10⁻³	0	1.945 ×10⁻⁵	0	1.945 ×10⁻⁷
0.8	0	2.904× 10⁻³	0	2.904× 10⁻⁵	0	2.904× 10⁻⁷
0.9	0	4.134×10⁻³	0	4.134×10⁻⁵	0	4.134×10⁻⁷
1.0	0	5.672× 10⁻³	0	5.672×10⁻⁵	0	5.672× 10⁻⁷

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

t	Exact	HAM	HWT Error	HWT
0	0.36787	0.36787	0.36787	0
0.1	0.40657	0.40657	0.41224	5.672 ×10⁻³
0.2	0.44932	0.44932	0.47202	2.268 ×10⁻²
0.3	0.49658	0.49658	0.54763	5.104 ×10⁻²
0.4	0.54881	0.54881	0.63956	9.075 ×10⁻²
0.5	0.60653	0.60653	0.74833	1.418×10⁻¹
0.6	0.67032	0.67032	0.87451	2.042×10⁻¹
0.7	0.74081	0.74081	1.01870	2.779 ×10⁻¹
0.8	0.81873	0.81873	1.18175	3.630×10⁻¹
0.9	0.90483	0.90483	1.36432	4.594×10⁻¹
1.0	1	1	1.56721	5.672×10⁻¹

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

x	Exact	HAM	HWT Error
0	1	1	1
0.1	0.90483	0.90483	0.90483
0.2	0.81873	0.81873	0.81873
0.3	0.74081	0.74081	0.74081
0.4	0.67032	0.67032	0.67032
0.5	0.60653	0.60653	0.60653
0.6	0.54881	0.54881	0.54881
0.7	0.49658	0.49658	0.49658
0.8	0.44932	0.44932	0.44932
0.9	0.40656	0.40656	0.40656
1.0	0.36787	0.36787	0.36787

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

x	Exact	HAM	HWT Error	HWT
0	2.71828	2.71828	2.71828	0
0.1	2.45960	2.45960	2.46021	5.672 ×10⁻⁴
0.2	2.22554	2.22554	2.23012	4.357 ×10⁻³
0.3	2.01375	2.01375	2.0291	1.531 ×10⁻²
0.4	1.82212	1.82212	1.8584	3.630×10⁻²
0.5	1.64872	1.64872	1.7196	7.090 ×10⁻²
0.6	1.49182	1.49182	1.6143	1.225 ×10⁻¹
0.7	1.34986	1.34986	1.5444	1.945 ×10⁻¹
0.8	1.22140	1.22140	1.5118	2.904 ×10⁻¹
0.9	1.10517	1.10517	1.5187	4.134×10⁻¹
1.0	1	1	1.5672	5.672×10⁻¹

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

x	Exact	HAM	HWT	HAM Error	HWT Error
0.5	0.541049	0.540974	0.54147	7.554×10⁻⁵	5.0×10⁻⁴
0.6	0.581622	0.581618	0.58234	4.062 ×10⁻⁶	7.2×10⁻⁴
0.7	0.614361	0.614361	0.61534	3.496×10⁻⁷	9.8 ×10⁻⁴
0.8	0.639926	0.639926	0.64121	2.917×10⁻⁷	1.28 ×10⁻³
0.9	0.658813	0.658813	0.66043	2.933 ×10⁻⁷	1.62 ×10⁻³
1.0	0.671397	0.671397	0.67340	2.929 ×10⁻⁷	2×10⁻³
1.1	0.677989	0.677989	0.68041	2.904 ×10⁻⁷	2.42 ×10⁻³
1.2	0.678866	0.678865	0.68175	2.838 ×10⁻⁷	2.88 ×10⁻³
1.3	0.674286	0.674289	0.67767	3.718×10⁻⁶	3.38 ×10⁻³
1.4	0.663534	0.664524	0.66844	9.9 ×10⁻⁴	3.92×10⁻³

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

x	Exact	HAM	HWT	HWT Error
0	1	1	1	0
0.1	0.90483	0.90483	0.90484	5.672 ×10⁻⁸
0.2	0.81873	0.81873	0.81873	9.075 ×10⁻⁷
0.3	0.74081	0.74081	0.74082	4.594 ×10⁻⁶
0.4	0.67032	0.67032	0.67057	1.452 ×10⁻⁵
0.5	0.60653	0.60653	0.60657	3.545 ×10⁻⁵
0.6	0.54881	0.54881	0.54889	7.350×10⁻⁵
0.7	0.49658	0.49658	0.49672	1.361 ×10⁻⁴
0.8	0.44932	0.44932	0.44956	2.323 ×10⁻⁴
0.9	0.40656	0.40656	0.40694	3.721 ×10⁻⁴
1.0	0.36787	0.36787	0.36845	5.672 ×10⁻⁴

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

Figures & Tables

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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_

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

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

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

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

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

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

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

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

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