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Year 2023, Volume: 11 Issue: 2, 141 - 147, 31.10.2023

Sandipan Dutta Sneha Gupta

Abstract

References

  • [1] Bischof, C.H., Pusch, G.O., Knoesel, R.: Sensitivity analysis of the mm5 weather model using automatic differentiation. Computers in Physics 10(6), 605–612 (1996)
  • [2] Bisi, C., De Martino, A., Winkelmann, J.: On a runge theorem over R3. Annali di Matematica 202(4), 1531–1556 (2023)
  • [3] Bisi, C., Winkelmann, J.: On a quaternionic Picard theorem. Proc. Amer. Math. Soc., Ser. B, 7, 106–117 (2020)
  • [4] Bisi, C., Winkelmann, J.: The harmonicity of slice regular functions. The Journal of Geometric Analysis 31(8), 7773–7811 (2021)
  • [5] Casanova, G.: Parabolic analytic functions. Adv. Appl. Clifford Algebras 9(2), 221–224 (1999)
  • [6] Catoni, F., Cannata, R., Catoni, V., Zampetti, P.: N-dimensional geometries generated by hypercomplex numbers. Adv. Appl. Clifford Algebr. 15(1), 1–25 (2005)
  • [7] Catoni, F., Cannata, R., Nichelatti, E.: The parabolic analytic functions and the derivative of real functions. Adv. Appl. Clifford Algebr. 14(2), 185–190 (2004)
  • [8] Fjelstad, P., Gal, S.G.: Two-dimensional geometries, topologies, trigonometries and physics generated by complex-type numbers. Adv. Appl. Clifford Algebras 11(1), 81–107 (2001)
  • [9] Gentili, G., Stoppato, C., Struppa, D.: Regular Functions of a Quaternionic Variable. Springer Monographs in Mathematics. Springer (2022)
  • [10] Hovland, P.D., et al: Sensitivity analysis and design optimization through automatic differentiation. J. Phys.: Conf. Ser. 16, 466–470 (2005)
  • [11] Kedem, G.: Automatic differentiation of computer programs. ACM Trans. Math. Software 6(2), 150–165 (1980)
  • [12] Mehmood, S., Ochs, P.: Automatic differentiation of some first-order methods in parametric optimization. In: Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, vol. 108, pp. 1584–1594. PMLR (2020)
  • [13] Pe˜nu nuri, F., Pe´on, R., Gonz´alez-S´anchez, D., Escalante Soberanis, M.A.: Dual numbers and automatic differentiation to efficiently compute velocities and accelerations. Acta Appl. Math. 170, 649–659 (2020)
  • [14] Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical recipes in Fortran 77: The art of scientific computing. Cambridge University Press (1992)
  • [15] Versteeg, H.K., Malalasekera, W.: An Introduction to Computational Fluid Dynamics: The Finite Volume Method. Pearson Education (2007)
  • [16] Yaglom, I.M.: A simple non-Euclidean geometry and its physical basis. Heidelberg Science Library. Springer-Verlag, New York-Heidelberg (1979). An elementary account of Galilean geometry and the Galilean principle of relativity, Translated from the Russian by Abe Shenitzer, With the editorial assistance of Basil Gordon.

Higher Order Real Derivatives Using Parabolic Analytic Functions

Year 2023, Volume: 11 Issue: 2, 141 - 147, 31.10.2023

Sandipan Dutta Sneha Gupta

Abstract

Amid the bidimensional hypercomplex numbers, parabolic numbers are defined as $\{z=x+\imath y:\; x,y\in \mathbb{R}, \imath^2=0, \imath\neq 0\}$. The analytic functions of a parabolic variable have been introduced as an analytic continuation of the real function of a real variable. Also, their algebraic property has already been discussed. This paper will show the $n$-th derivative of the real functions using parabolic numbers to further generalize the automatic differentiation. Also, we shall show some of the applications of it.

Keywords

Parabolic Analytic functions Dual number Higher order derivative automatic differentiation Hypercomplex numbers.

References

  • [1] Bischof, C.H., Pusch, G.O., Knoesel, R.: Sensitivity analysis of the mm5 weather model using automatic differentiation. Computers in Physics 10(6), 605–612 (1996)
  • [2] Bisi, C., De Martino, A., Winkelmann, J.: On a runge theorem over R3. Annali di Matematica 202(4), 1531–1556 (2023)
  • [3] Bisi, C., Winkelmann, J.: On a quaternionic Picard theorem. Proc. Amer. Math. Soc., Ser. B, 7, 106–117 (2020)
  • [4] Bisi, C., Winkelmann, J.: The harmonicity of slice regular functions. The Journal of Geometric Analysis 31(8), 7773–7811 (2021)
  • [5] Casanova, G.: Parabolic analytic functions. Adv. Appl. Clifford Algebras 9(2), 221–224 (1999)
  • [6] Catoni, F., Cannata, R., Catoni, V., Zampetti, P.: N-dimensional geometries generated by hypercomplex numbers. Adv. Appl. Clifford Algebr. 15(1), 1–25 (2005)
  • [7] Catoni, F., Cannata, R., Nichelatti, E.: The parabolic analytic functions and the derivative of real functions. Adv. Appl. Clifford Algebr. 14(2), 185–190 (2004)
  • [8] Fjelstad, P., Gal, S.G.: Two-dimensional geometries, topologies, trigonometries and physics generated by complex-type numbers. Adv. Appl. Clifford Algebras 11(1), 81–107 (2001)
  • [9] Gentili, G., Stoppato, C., Struppa, D.: Regular Functions of a Quaternionic Variable. Springer Monographs in Mathematics. Springer (2022)
  • [10] Hovland, P.D., et al: Sensitivity analysis and design optimization through automatic differentiation. J. Phys.: Conf. Ser. 16, 466–470 (2005)
  • [11] Kedem, G.: Automatic differentiation of computer programs. ACM Trans. Math. Software 6(2), 150–165 (1980)
  • [12] Mehmood, S., Ochs, P.: Automatic differentiation of some first-order methods in parametric optimization. In: Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, vol. 108, pp. 1584–1594. PMLR (2020)
  • [13] Pe˜nu nuri, F., Pe´on, R., Gonz´alez-S´anchez, D., Escalante Soberanis, M.A.: Dual numbers and automatic differentiation to efficiently compute velocities and accelerations. Acta Appl. Math. 170, 649–659 (2020)
  • [14] Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical recipes in Fortran 77: The art of scientific computing. Cambridge University Press (1992)
  • [15] Versteeg, H.K., Malalasekera, W.: An Introduction to Computational Fluid Dynamics: The Finite Volume Method. Pearson Education (2007)
  • [16] Yaglom, I.M.: A simple non-Euclidean geometry and its physical basis. Heidelberg Science Library. Springer-Verlag, New York-Heidelberg (1979). An elementary account of Galilean geometry and the Galilean principle of relativity, Translated from the Russian by Abe Shenitzer, With the editorial assistance of Basil Gordon.
There are 16 citations in total.

Details

Primary Language English
Subjects Complex Systems in Mathematics
Journal Section Articles
Authors

Sandipan Dutta

Sneha Gupta

Publication Date October 31, 2023
Submission Date July 18, 2023
Acceptance Date October 21, 2023
Published in Issue Year 2023 Volume: 11 Issue: 2

Cite

APA Dutta, S., & Gupta, S. (2023). Higher Order Real Derivatives Using Parabolic Analytic Functions. Konuralp Journal of Mathematics, 11(2), 141-147.
AMA Dutta S, Gupta S. Higher Order Real Derivatives Using Parabolic Analytic Functions. Konuralp J. Math. October 2023;11(2):141-147.
Chicago Dutta, Sandipan, and Sneha Gupta. “Higher Order Real Derivatives Using Parabolic Analytic Functions”. Konuralp Journal of Mathematics 11, no. 2 (October 2023): 141-47.
EndNote Dutta S, Gupta S (October 1, 2023) Higher Order Real Derivatives Using Parabolic Analytic Functions. Konuralp Journal of Mathematics 11 2 141–147.
IEEE S. Dutta and S. Gupta, “Higher Order Real Derivatives Using Parabolic Analytic Functions”, Konuralp J. Math., vol. 11, no. 2, pp. 141–147, 2023.
ISNAD Dutta, Sandipan - Gupta, Sneha. “Higher Order Real Derivatives Using Parabolic Analytic Functions”. Konuralp Journal of Mathematics 11/2 (October 2023), 141-147.
JAMA Dutta S, Gupta S. Higher Order Real Derivatives Using Parabolic Analytic Functions. Konuralp J. Math. 2023;11:141–147.
MLA Dutta, Sandipan and Sneha Gupta. “Higher Order Real Derivatives Using Parabolic Analytic Functions”. Konuralp Journal of Mathematics, vol. 11, no. 2, 2023, pp. 141-7.
Vancouver Dutta S, Gupta S. Higher Order Real Derivatives Using Parabolic Analytic Functions. Konuralp J. Math. 2023;11(2):141-7.
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