A Novel Formulation of the Fractional Derivative with the Order and without the Singular Kernel
Abstract
:1. Introduction
2. New Definition of Fractional Derivative
- Using the definition of the new fractional derivative and the properties of the integral, this relationship can be easily proven.
- Using Equation (2) and since ,the following relationship can be obtained:
- Using the definition of the new derivative and since , then thus, Equation (11) is true. □
- By definition of new fractional derivative and integral of the exponential function, it is possible to prove the following relationship:
- Relationship No. 2 can easily be proven using the new derivative definition and the integral of the sine function:
- Similarly, relationship No. 3 can be proved. □
- From Equations (2) and (20), we obtain:
- Using Equations (2) and (20), we obtain:
3. Integral Transforms of the New Derivative
4. Initial Value Problems with New Derivative
5. Illustrative Examples
0.2143 | 0.8016 | 0.8024 | 0.7629 | 0.7367 | 0.8071 | 0.0055 | 0.0047 | 0.0442 | 0.0704 |
0.4286 | 0.6482 | 0.6473 | 0.6402 | 0.6165 | 0.6514 | 0.0032 | 0.0041 | 0.0112 | 0.0349 |
0.6429 | 0.5241 | 0.5253 | 0.5373 | 0.5226 | 0.5258 | 0.0016 | 0.0005 | 0.0115 | 0.0032 |
0.8571 | 0.4238 | 0.4289 | 0.4509 | 0.4469 | 0.4244 | 0.0005 | 0.0046 | 0.0265 | 0.0226 |
1.0714 | 0.3427 | 0.3525 | 0.3784 | 0.3851 | 0.3425 | 0.0002 | 0.0100 | 0.0358 | 0.0426 |
1.2857 | 0.2771 | 0.2917 | 0.3175 | 0.3339 | 0.2765 | 0.0007 | 0.0152 | 0.0411 | 0.0575 |
1.5000 | 0.2241 | 0.2431 | 0.2664 | 0.2913 | 0.2231 | 0.0010 | 0.0200 | 0.0433 | 0.0681 |
1.7143 | 0.1812 | 0.2041 | 0.2236 | 0.2554 | 0.1801 | 0.0011 | 0.0240 | 0.0435 | 0.0754 |
1.9286 | 0.1465 | 0.1726 | 0.1876 | 0.2252 | 0.1454 | 0.0012 | 0.0273 | 0.0423 | 0.0798 |
2.1429 | 0.1185 | 0.1471 | 0.1575 | 0.1994 | 0.1173 | 0.0012 | 0.0298 | 0.0401 | 0.0821 |
2.3571 | 0.0958 | 0.1264 | 0.1321 | 0.1775 | 0.0947 | 0.0011 | 0.0317 | 0.0375 | 0.0828 |
2.5714 | 0.0775 | 0.1094 | 0.1109 | 0.1586 | 0.0764 | 0.0010 | 0.0330 | 0.0345 | 0.0822 |
2.7857 | 0.0626 | 0.0954 | 0.0931 | 0.1424 | 0.0617 | 0.0010 | 0.0338 | 0.0314 | 0.0808 |
3.0000 | 0.0507 | 0.0839 | 0.0781 | 0.1284 | 0.0498 | 0.0009 | 0.0341 | 0.0283 | 0.0786 |
0.0714 | 0.0714 | 0.0667 | 0.0667 | 0.0667 | 0.0667 | 0.0667 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
0.1429 | 0.1429 | 0.1254 | 0.1254 | 0.1254 | 0.1254 | 0.1250 | 0.0004 | 0.0004 | 0.0004 | 0.0004 |
0.2143 | 0.2143 | 0.1782 | 0.1782 | 0.1782 | 0.1782 | 0.1765 | 0.0017 | 0.0017 | 0.0017 | 0.0017 |
0.2857 | 0.2857 | 0.2274 | 0.2274 | 0.2274 | 0.2274 | 0.2222 | 0.0052 | 0.0052 | 0.0052 | 0.0052 |
0.3571 | 0.3571 | 0.2751 | 0.2751 | 0.2751 | 0.2751 | 0.2632 | 0.0120 | 0.0120 | 0.0120 | 0.0120 |
0.4286 | 0.4286 | 0.3236 | 0.3236 | 0.3236 | 0.3236 | 0.3000 | 0.0236 | 0.0236 | 0.0236 | 0.0236 |
0.5000 | 0.5000 | 0.3750 | 0.3750 | 0.3750 | 0.3750 | 0.3333 | 0.0417 | 0.0417 | 0.0417 | 0.0417 |
0.5714 | 0.5714 | 0.4315 | 0.4315 | 0.4315 | 0.4315 | 0.3636 | 0.0679 | 0.0679 | 0.0679 | 0.0679 |
0.6429 | 0.6429 | 0.4953 | 0.4953 | 0.4953 | 0.4953 | 0.3913 | 0.1040 | 0.1040 | 0.1040 | 0.1040 |
0.7143 | 0.7143 | 0.5685 | 0.5685 | 0.5685 | 0.5685 | 0.4167 | 0.1518 | 0.1518 | 0.1518 | 0.1518 |
0.7857 | 0.7857 | 0.6534 | 0.6534 | 0.6534 | 0.6534 | 0.4400 | 0.2134 | 0.2134 | 0.2134 | 0.2134 |
0.8571 | 0.8571 | 0.7522 | 0.7522 | 0.7522 | 0.7522 | 0.4615 | 0.2906 | 0.2906 | 0.2906 | 0.2906 |
0.9286 | 0.9286 | 0.8670 | 0.8670 | 0.8670 | 0.8670 | 0.4815 | 0.3855 | 0.3855 | 0.3855 | 0.3855 |
1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 0.5000 | 0.5000 | 0.5000 | 0.5000 | 0.5000 |
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Jassim, H.K.; Hussein, M.A.
A Novel Formulation of the Fractional Derivative with the Order
Jassim HK, Hussein MA.
A Novel Formulation of the Fractional Derivative with the Order
Jassim, Hassan Kamil, and Mohammed A. Hussein.
2022. "A Novel Formulation of the Fractional Derivative with the Order
Jassim, H. K., & Hussein, M. A.
(2022). A Novel Formulation of the Fractional Derivative with the Order