Reduction to a Fredholm integral equation and numerical solution of the inverse Cauchy problem for the Schrödinger-Pauli equation

Yusif Gasimov; Abdeljalil Nachaoui; Aynura Aliyeva

doi:10.2478/ijmce-2026-0015

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Reduction to a Fredholm integral equation and numerical solution of the inverse Cauchy problem for the Schrödinger-Pauli equation

International Journal of Mathematics and Computer in Engineering

AHEAD OF PRINT

By: Yusif Gasimov, Abdeljalil Nachaoui and Aynura Aliyeva

Open Access

|Jun 2026

Figures & Tables

L-curve for m = 1, J = 3, showing the trade-off between solution norm and residual.

Boundary Trace Reconstruction for m = 1, δ = 0, J = 3 and Detail.

Exact (left) and reconstructed (right) solutions for m = 1.

Error profile along y at x = 0.5 (m = 1, δ = 0).

Boundary trace reconstruction for m = 1 with different noise levels, J = 3.

Boundary trace reconstruction for the multi-mode solution, δ = 0, J = 3.

Boundary trace reconstruction for the multi-mode solution with different noise levels, J = 3.

Convergence study for m = 1, δ = 0_

J	M = 2^J	Relative Error ℰ	Convergence Rate	Error Reduction
1	2	6.21e-02	–	–
2	4	3.15e-02	0.98	1.97
3	8	1.58e-02	1.00	1.99
4	16	7.92e-03	1.00	2.00
5	32	3.96e-03	1.00	2.00
6	64	1.98e-03	1.00	2.00

Comparison of error increase factors: Single mode (m = 1) vs Multi-mode_

Noise Level	Error Increase Factor	Error Increase Factor	Ratio
δ	Single mode (m = 1)	Multi-mode	(Multi/Single)
0.001	1.13	1.02	0.90
0.005	1.63	1.09	0.67
0.010	2.19	1.23	0.56
0.050	5.56	2.38	0.43
0.100	8.88	3.89	0.44

Values of γn and βn for selected n_

n	γ_n	β_n	n	γ_n	β_n
1	4.3973	0.0251	7	22.4306	3.48 × 10⁻¹¹
2	7.0963	0.0009	8	25.7158	1.09 × 10⁻¹²
3	9.9713	3.16 × 10⁻⁵	9	29.0538	3.42 × 10⁻¹⁴
4	12.9614	1.06 × 10⁻⁶	10	32.4397	1.07 × 10⁻¹⁵
5	16.0442	3.44 × 10⁻⁸	15	49.9741	3.10 × 10⁻²³
6	19.2042	1.10 × 10⁻⁹	20	68.3345	8.70 × 10⁻³¹

Performance comparison and error increase factors for different noise levels m = 1 J = 3

Noise Level δ	Relative Error ℰ(δ)	Optimal α	Error Increase Factor	Noise-to-Error Ratio	α Increase Factor
0.000	0.016	0.001	1.00	–	1.00
0.001	0.018	0.001	1.13	0.056	1.00
0.005	0.026	0.003	1.63	0.200	3.00
0.010	0.035	0.005	2.19	0.286	5.00
0.050	0.089	0.018	5.56	0.562	18.00
0.100	0.142	0.032	8.88	0.704	32.00

Performance comparison and error increase factors for multi-mode solution with different noise levels, J = 3_

Noise Level δ	Relative Error ℰ(δ)	Optimal α	Error Increase Factor	Noise-to-Error Ratio	α Increase Factor
0.000	0.047	0.005	1.00	–	1.00
0.001	0.048	0.005	1.02	0.021	1.00
0.005	0.051	0.007	1.09	0.098	1.40
0.010	0.058	0.010	1.23	0.172	2.00
0.050	0.112	0.032	2.38	0.446	6.40
0.100	0.183	0.056	3.89	0.546	11.20

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DOI: https://doi.org/10.2478/ijmce-2026-0015 | Journal eISSN: 2956-7068

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Language: English

Submitted on: Jan 12, 2026

Accepted on: Feb 24, 2026

Published on: Jun 2, 2026

Published by: Harran University

In partnership with: Paradigm Publishing Services

Publication frequency: 2 issues per year

Keywords:

Inverse Cauchy problem,

Schrödinger-Pauli equation,

Fredholm integral equation,

Haar wavelet,

collocation scheme

Related subjects:

Computer sciences,

Computer sciences, other,

Engineering,

Introductions and overviews,

Physics,

© 2026 Yusif Gasimov, Abdeljalil Nachaoui, Aynura Aliyeva, published by Harran University
This work is licensed under the Creative Commons Attribution 4.0 License.

AHEAD OF PRINT

Reduction to a Fredholm integral equation and numerical solution of the inverse Cauchy problem for the Schrödinger-Pauli equation

Figures & Tables

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Convergence study for m = 1, δ = 0_

Comparison of error increase factors: Single mode (m = 1) vs Multi-mode_

Values of γn and βn for selected n_

Performance comparison and error increase factors for different noise levels m = 1 J = 3

Performance comparison and error increase factors for multi-mode solution with different noise levels, J = 3_

Paradigm

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