Main Article Content
The Double Sadik Transform (DST) represents a generalized double integral transform that has emerged as a highly effective analytical technique for solving numerous scientific problems. This study aims to investigate the DST applied to elementary functions and explore its notable properties, including its duality with the Double Laplace Transform and its capability to transform shifting functions, periodic functions, and convolution functions. Furthermore, the DST methodology is employed to resolve prominent linear fractional Caputo partial differential equations with known solutions commonly encountered in diverse mathematical models. The obtained outcomes are expressed in exact closed form, with the most precise results articulated through the Mittag-Leffler function. These results serve to validate the effectiveness and efficiency of the DST approach, establishing it as a valuable tool for addressing scientific problems involving fractional calculus.
- H. Jafari, V. Daftardar-Gejji, Solving a System of Nonlinear Fractional Differential Equations Using Adomian Decomposition, J. Comput. Appl. Math. 196 (2006), 644–651. https://doi.org/10.1016/j.cam.2005.10.017.
- A.A. Hemeda, Modified Homotopy Perturbation Method for Solving Fractional Differential Equations, J. Appl. Math. 2014 (2014), 594245. https://doi.org/10.1155/2014/594245.
- A.S.V. Ravi Kanth, K. Aruna, Solution of Fractional Third-Order Dispersive Partial Differential Equations, Egypt. J. Basic Appl. Sci. 2 (2015), 190–199. https://doi.org/10.1016/j.ejbas.2015.02.002.
- M. Dehghan, J. Manafian, A. Saadatmandi, The Solution of the Linear Fractional Partial Differential Equations Using the Homotopy Analysis Method, Z. Naturforsch. A. 65 (2010), 935–949. https://doi.org/10.1515/zna-2010-1106.
- V. Turut, N. Güzel, On Solving Partial Differential Equations of Fractional Order by Using the Variational Iteration Method and Multivariate Padé Approximations, Eur. J. Pure Appl. Math. 6 (2013), 147–171.
- L. Kexue, P. Jigen, Laplace Transform and Fractional Differential Equations, Appl. Math. Lett. 24 (2011), 2019– 2023. https://doi.org/10.1016/j.aml.2011.05.035.
- M. Bahri, S.A. Abdul Karim, Fractional Fourier Transform: Main Properties and Inequalities, Mathematics. 11 (2023), 1234. https://doi.org/10.3390/math11051234.
- M. Akram, T. Ihsan, Solving Pythagorean Fuzzy Partial Fractional Diffusion Model Using the Laplace and Fourier Transforms, Granul. Comput. 8 (2022), 689–707. https://doi.org/10.1007/s41066-022-00349-8.
- P. Hammachukiattikul, A. Mohanapriya, A. Ganesh, G. Rajchakit, V. Govindan, N. Gunasekaran, C.P. Lim, A Study on Fractional Differential Equations Using the Fractional Fourier Transform, Adv. Differ Equ. 2020 (2020), 691. https://doi.org/10.1186/s13662-020-03148-0.
- S. Tuluce Demiray, H. Bulut, F.B.M. Belgacem, Sumudu Transform Method for Analytical Solutions of Fractional Type Ordinary Differential Equations, Math. Probl. Eng. 2015 (2015), 131690. https://doi.org/10.1155/2015/131690.
- M.Z. Mohamed, M. Yousif, A.E. Hamza, Solving Nonlinear Fractional Partial Differential Equations Using the Elzaki Transform Method and the Homotopy Perturbation Method, Abstr. Appl. Anal. 2022 (2022), 4743234. https://doi.org/10.1155/2022/4743234.
- T.G. Thange, A.R. Gade, On Aboodh Transform for Fractional Differential Operator, Malaya J. Mat. 8 (2020), 225–229. https://doi.org/10.26637/mjm0801/0038.
- M.A. Awuya, D. Subasi, Aboodh Transform Iterative Method for Solving Fractional Partial Differential Equation with Mittag–Leffler Kernel, Symmetry. 13 (2021), 2055. https://doi.org/10.3390/sym13112055.
- S. Butera, M. Di Paola, Fractional Differential Equations Solved by Using Mellin Transform, Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 2220–2227. https://doi.org/10.1016/j.cnsns.2013.11.022.
- Y. Luchko, V. Kiryakova, The Mellin Integral Transform in Fractional Calculus, Fract. Calc. Appl. Anal. 16 (2013), 405–430. https://doi.org/10.2478/s13540-013-0025-8.
- P. Pue-On, The Modified Sadik Decomposition Method to Solve a System of Nonlinear Fractional Volterra IntegroDifferential Equations of Convolution type, WSEAS Trans. Math. 20 (2021), 335–343. https://doi.org/10.37394/23206.2021.20.34.
- S.S. Redhwan, S.L. Shaikh, M.S. Abdo, Some Properties of Sadik Transform and Its Applications of FractionalOrder Dynamical Systems in Control Theory, Adv. Theory Nonlinear Anal. Appl. 4 (2020), 51–66. https://doi.org/10.31197/atnaa.647503.
- M.M. Khader, N.H. Sweilam, On the Approximate Solutions for System of Fractional Integro-Differential Equations Using Chebyshev Pseudo-Spectral Method, Appl. Math. Model. 37 (2013), 9819–9828. https://doi.org/10.1016/j.apm.2013.06.010.
- A.M.O. Anwar, F. Jarad, D. Baleanu, F. Ayaz, Frational Caputo Heat Equation Within the Double Laplace Transform, Rom. J. Phys. 58 (2013), 15–22.
- L. Debnath, The Double Laplace Transforms and Their Properties with Applications to Functional, Integral and Partial Differential Equations, Int. J. Appl. Comput. Math. 2 (2015), 223–241. https://doi.org/10.1007/s40819-015-0057-3.
- M.A. Hassan, T.M. Elzaki, Double Elzaki Transform Decomposition Method for Solving Non-Linear Partial Differential Equations, J. Appl. Math. Phys. 08 (2020), 1463–1471. https://doi.org/10.4236/jamp.2020.88112.
- H. Eltayeb, A. Kilicman, On Double Sumudu Transform and Double Laplace Transform, Malays. J. Math. Sci. 4 (2010), 17–30.
- S.M. Sonawane, S.B. Kiwne, Double Kamal transforms: Properties and Applications, Int. J. Appl. Sci. Comput. 6 (2019), 1727–1739.
- S. Alfaqeih, E. Misirli, On Double Shehu Transform and Its Properties With Applications, Int. J. Anal. Appl. 18 (2020), 381–395. https://doi.org/10.28924/2291-8639-18-2020-381.
- A.K. Sedeeg, Zahra.I. Mahamoud, R. Saadeh, Using Double Integral Transform (Laplace-ARA Transform) in Solving Partial Differential Equations, Symmetry. 14 (2022), 2418. https://doi.org/10.3390/sym14112418.
- P. Pue-on, The Exact Solutions of the Space and Time Fractional Telegraph Equations by the Double Sadik Transform Method, Math. Stat. 10 (2022), 995–1004. https://doi.org/10.13189/ms.2022.100511.
- S.L. Shaikh, Introducing a New Integral Transform Sadik Transform, Amer. Int. J. Res. Sci. Technol. Eng. Math. 22 (2018), 100–102. https://doi.org/10.13140/RG.2.2.25805.08161.
- Y. Singh, On Some Theorems and Applications of Double Sadik Transform, Compliance Eng. J. 10 (2019), 164–174.
- R.R. Dhunde, G.L. Waghmare, Solutions of Some Linear Fractional Partial Differential Equations in Mathematical Physics, J. Indian Math. Soc. 85 (2018), 313–327. https://doi.org/10.18311/jims/2018/20144.