An Approximate Analytical View of Fractional Physical Models in the Frame of the Caputo Operator
Abstract
1. Introduction
2. Preliminaries
3. Construction of HPTM
4. Construction of YTDM
5. Convergence Analysis
6. Error Estimation
7. Test Examples
7.1. Example
- Application of HPTM
- Application of YTDM
7.2. Example
- Application of HPTM
- Application of YTDM
7.3. Example
- Application of HPTM
- Application of YTDM
8. Physical Interpretation of Results
9. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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(Appro) | (Exact) | ||||
---|---|---|---|---|---|
0.0 | 0.8412492045 | 0.8412821476 | 0.8413116951 | 0.8413381923 | 0.8413382266 |
0.1 | 0.8347832441 | 0.8346459868 | 0.8345135197 | 0.8343858026 | 0.8343849486 |
0.2 | 0.8089819894 | 0.8086852743 | 0.8084011555 | 0.8081292105 | 0.8081276073 |
0.3 | 0.7658214625 | 0.7653876267 | 0.7649732281 | 0.7645774965 | 0.7645753904 |
0.4 | 0.7084342409 | 0.7078937677 | 0.7073781753 | 0.7068864044 | 0.7068840866 |
0.5 | 0.6406493580 | 0.6400366044 | 0.6394525467 | 0.6388959115 | 0.6388936545 |
0.6 | 0.5664939123 | 0.5658429627 | 0.5652228708 | 0.5646322278 | 0.5646302394 |
0.7 | 0.4897564694 | 0.4890978532 | 0.4884707464 | 0.4878736802 | 0.4878720841 |
0.8 | 0.4136762461 | 0.4130348823 | 0.4124244238 | 0.4118434074 | 0.4118422460 |
0.9 | 0.3407780718 | 0.3401724392 | 0.3395961584 | 0.3390478223 | 0.3390470757 |
1.0 | 0.2728373544 | 0.2722796467 | 0.2717490940 | 0.2712443819 | 0.2712439912 |
Exact Solution | Our Methods Solution | Our Methods Error | LRPSM Error | HASTM Error | NTDM Error | ||
---|---|---|---|---|---|---|---|
0.5 | 0 | 0.6280127585 | 0.6280127585 | 0.0000000000 | 0.0000000000 | 0.0000000000 | 0.0000000000 |
0.01 | 0.6388936545 | 0.6388959115 | 2.256967 | 2.26691 | 3.51211 | 6.63723 | |
0.02 | 0.6496327491 | 0.6496508562 | 1.810706 | 1.81241 | 2.81021 | 7.7452 | |
0.03 | 0.6602163246 | 0.6602775927 | 6.126800 | 6.12895 | 9.48603 | 1.38205 | |
0.04 | 0.6706305549 | 0.6707761209 | 1.455658 | 1.45589 | 2.24888 | 1.17 | |
0.05 | 0.6808615377 | 0.6811464403 | 2.849025 | 2.84925 | 4.39294 | 1.8875 | |
1.0 | 0 | 0.2615393671 | 0.2615393671 | 0.0000000000 | 0.0000000000 | 0.0000000000 | 0.0000000000 |
0.01 | 0.2712439912 | 0.2712443819 | 3.906838 | 3.99819 | 1.41429 | 7.0112 | |
0.02 | 0.2810871794 | 0.2810904023 | 3.222867 | 3.23987 | 2.24888 | 2.78788 | |
0.03 | 0.2910662174 | 0.2910774282 | 1.121086 | 1.12342 | 3.81193 | 1.2331 | |
0.04 | 0.3011780846 | 0.3012054598 | 2.737526 | 2.74038 | 9.02767 | 1.1006 | |
0.05 | 0.3114194433 | 0.3114744970 | 5.505375 | 5.50858 | 1.76163 | 1.0116 |
(Appro) | (Exact) | ||||
---|---|---|---|---|---|
0.0 | 0.8412492045 | 0.8412821476 | 0.8413116951 | 0.8413381923 | 0.8413382266 |
0.1 | 0.8347832441 | 0.8346459868 | 0.8345135197 | 0.8343858026 | 0.8343849486 |
0.2 | 0.8089819894 | 0.8086852743 | 0.8084011555 | 0.8081292105 | 0.8081276073 |
0.3 | 0.7658214625 | 0.7653876267 | 0.7649732281 | 0.7645774965 | 0.7645753904 |
0.4 | 0.7084342409 | 0.7078937677 | 0.7073781753 | 0.7068864044 | 0.7068840866 |
0.5 | 0.6406493580 | 0.6400366044 | 0.6394525467 | 0.6388959115 | 0.6388936545 |
0.6 | 0.5664939123 | 0.5658429627 | 0.5652228708 | 0.5646322278 | 0.5646302394 |
0.7 | 0.4897564694 | 0.4890978532 | 0.4884707464 | 0.4878736802 | 0.4878720841 |
0.8 | 0.4136762461 | 0.4130348823 | 0.4124244238 | 0.4118434074 | 0.4118422460 |
0.9 | 0.3407780718 | 0.3401724392 | 0.3395961584 | 0.3390478223 | 0.3390470757 |
1.0 | 0.2728373544 | 0.2722796467 | 0.2717490940 | 0.2712443819 | 0.2712439912 |
0.0 | 0.9602125488 | 0.9643569874 | 0.9680748874 | 0.9714095642 | 0.9744000000 |
0.1 | 0.9904392302 | 0.9925362180 | 0.9943131706 | 0.9958075547 | 0.9970528593 |
0.2 | 1.0014573480 | 1.0013532710 | 1.0010595720 | 1.0006050540 | 1.0000151990 |
0.3 | 0.9922549156 | 0.9899921951 | 0.9876807940 | 0.9853390759 | 0.9829829438 |
0.4 | 0.9634794619 | 0.9592829050 | 0.9551749023 | 0.9511628260 | 0.9472525522 |
0.5 | 0.9173351619 | 0.9115676279 | 0.9060098467 | 0.9006588122 | 0.8955109712 |
0.6 | 0.8572164277 | 0.8503175576 | 0.8437251030 | 0.8374271840 | 0.8314122178 |
0.7 | 0.7871841277 | 0.7796080070 | 0.7724064574 | 0.7655607516 | 0.7590531407 |
0.8 | 0.7114189494 | 0.7035832173 | 0.6961618533 | 0.6891314651 | 0.6824701346 |
0.9 | 0.6337652853 | 0.6260180752 | 0.6186998385 | 0.6117845694 | 0.6052480500 |
1.0 | 0.5574285929 | 0.5500337403 | 0.5430621488 | 0.5364869364 | 0.5302831645 |
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AlBaidani, M.M.; Ganie, A.H.; Khan, A.; Aljuaydi, F. An Approximate Analytical View of Fractional Physical Models in the Frame of the Caputo Operator. Fractal Fract. 2025, 9, 199. https://doi.org/10.3390/fractalfract9040199
AlBaidani MM, Ganie AH, Khan A, Aljuaydi F. An Approximate Analytical View of Fractional Physical Models in the Frame of the Caputo Operator. Fractal and Fractional. 2025; 9(4):199. https://doi.org/10.3390/fractalfract9040199
Chicago/Turabian StyleAlBaidani, Mashael M., Abdul Hamid Ganie, Adnan Khan, and Fahad Aljuaydi. 2025. "An Approximate Analytical View of Fractional Physical Models in the Frame of the Caputo Operator" Fractal and Fractional 9, no. 4: 199. https://doi.org/10.3390/fractalfract9040199
APA StyleAlBaidani, M. M., Ganie, A. H., Khan, A., & Aljuaydi, F. (2025). An Approximate Analytical View of Fractional Physical Models in the Frame of the Caputo Operator. Fractal and Fractional, 9(4), 199. https://doi.org/10.3390/fractalfract9040199