Document Type : Original Article


1 University of Science & Technology, College of Engineering, Sudan

2 Al-Mughtaribeen University, College of Engineering, Department of General Sciences, Sudan



H. Jafari proposed a new integral transform recently, namely, the Jafari transform, which covered all classes of integral transforms in the class of Laplace transform, such as Laplace, Sumudu, Elzaki, Aboodh, natural, Shehu transformation, etc. In this paper, we utilize a semi-analytical technique, namely the Jafari variational iteration method, abbreviated JVIM, and we apply this technique to resolve one-dimensional diffusion equations with fractional order type using the Caputo fractional derivative. The results are compared with the homotopy analysis Shehu transform method (HASTM). Also, the results show the suggested algorithm is efficient, accurate and a powerful technique for solving a wide variety of linear and non-linear problems arising in various scientific areas.


[1] A. S. Abedl-Rady, S. Z. Rida, A. A. M. Arafa, and H. R. Abedl-Rahim, New technique for solving fractional physical equations, Sch. J. Phys. Math. Stat., 3(3) (2016), 110–116.
[2] K.S. Aboodh, The new integral transform aboodh transform, Glob. J. Pure Appl. Math., 9(1) (2013), 35–43.
[3] K. S. Aboodh and A. Ahmed, On the application of homotopy analysis method to fractional differential equations, J. Faculty Sci. Technol. 7 (2020), 1–18.
[4] S.A.P. Ahmadi, H. Hosseinzadeh, and A.Y. Cherati, A new integral transform for solving higher order linear ordinary differential equations, Nonlinear Dyn. Syst. Theory, 19(2) (2019), 243–252.
[5] F.A. Alawad, E.A. Yousif, and A.I. Arbab, A new technique of Laplace variational iteration method for solving space-time fractional telegraph equations, Int. J. Differ. Equ., 2013 (2013).
[6] H. Anac, A local fractional Elzaki transform decomposition method for the nonlinear system of local fractional partial differential equations, Fractal Fract. 6(3) (2022), 167.
[7] S. Cetinkaya, A. Demir, and H.K. Sevindir, Solution of space-time-fractional problem by Shehu variational iteration method, Adv. Math. Phys., 2021 (2021), 1–8.
[8] M. H. Cherif and D. Ziane, Variational iteration method combined with new transform to solve fractional partial differential equations, Univer. J. Math. Appl., 1(2) (2018), 113–120.
[9] H. Eltayeb, A. Kiliman, and B. Fisher, A new integral transform and associated distributions, Integral Transforms Special Funct. 21(5) (2010), 367–379.
[10] T.M. Elzaki, The new integral transform Elzaki transform, Glob. J. Pure Appl. Math., 7(1) (2011), 57–64.
[11] K. M. Furati and N. Tatar, An existence result to a nonlocal fractional problem, J. Fract. Calc., 26 (2004), 43–51.
[12] D. D. Ganji, H. Tari, and M. B. Jooybari, Variational iteration method and homotopy perturbation method for nonlinear evolution equations, Comput. Math. Appl., 54(7–8) (2007), 1018–1027.
[13] J. H. He, Homotopy perturbation technique, Comput. Meth. Appl. Mech. Engin., 178(3–4) (1999), 257–262.
[14] J. H. He, Variational iteration method-a kind of non-linear analytical technique: some examples, Int. J. Nonlinear Mech., 34(4) (1999), 699–708.
[15] J. H. He and X. H. Wu, Variational iteration method: New development and applications, Comput. Math. Appl., 54(7-8) (2007), 881–894.
[16] H. Jafari, A new general integral transform for solving integral equations, J. Adv. Res., 32 (2021), 133–138.
[17] H. Jafari and V. Daftardar-Gejji, Solving linear and nonlinear fractional diffusion and wave equations by adomian decomposition, J. Appl. Math. Comput., 180 (2006), 488–497.
[18] H. Jafari and S. Momani, Solving fractional diffusion and wave equations by modified homotopy perturbation method, Phys. Lett. A, 370(5–6) (2007), 388–396.
[19] H. Jafari and S. Seifi, Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation, Commun. Nonlinear Sci. Numer. Simul., 14(5) (2009), 2006–2012.
[20] H. M. Jaradat, Dynamic behavior of traveling wave solutions for a class for the time-space coupled fractional KdV system with time-dependent coefficients, Italian J. Pure Appl. Math., 36 (2016), 945–958.
[21] H. K. Jassim and S. A. Khafif, SVIM for solving Burger’s and coupled Burger’s equations of Fractional Order, Progr. Fract. Differ. Appl. 7(1) (2021), 73-78.
[22] H. Kamal and A Sedeeg, The new integral transform Kamal transform, Adv. Theor. Appl. Math., 11(4) (2016), 451–458.
[23] Q. D. Katatbeh and F. B. M. Belgacem, Applications of the Sumudu transform to fractional differential equations, Nonlinear Stud., 18 (2011), 99–112.
[24] A. Khalouta, A new general integral transform for solving Caputo fractional-order differential equations, Int. J. Nonlinear Anal. Appl., 14(1) (2023), 67–78.
[25] Z. H. Khan and W. A. Khan, N-transform properties and applications, NUST J. Eng. Sci., 1(1) (2008), 127–33.
[26] A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and application of fractional differential equations, Elsevier, North-Holland, 2006.
[27] H. Kim, On the form and properties of an integral transform with strength in integral transforms, Far East J. Math. Sci., 102(11) (2017), 2831–2844.
[28] H. Kim, The intrinsic structure and properties of Laplace-typed integral transforms, Math. Prob. Eng., 2017 (2017), Article ID 1762729, 8 pages.
[29] D. Kumar, J. Singh, and S. Kumar, Numerical computation of fractional multidimensional diffusion equations by using a modified homotopy perturbation method, J. Assoc. Arab Univer. Basic Appl. Sci., 17(1) (2015), 20–26.
[30] M.M.A. Mahgoub, The new integral transform sawi transform, Adv. Oretic. Appl. Math., 14(1) (2019), 81–87.
[31] S. Maitama and I. Abdullahi, A new analytical method for solving linear and nonlinear fractional partial differential equations, Progr. Fract. Differ. Appl., 2 (2016), 225–247.
[32] S. Maitama, M. S. Rawashdeh, and S. Sulaiman, An analytical method for solving linear and nonlinear Schr¨odinger equations, Palestine J. Math., 6 (2017), 59–67.
[33] S. Maitama and W. Zhao, New homotopy analysis transform method for solving multidimensional fractional diffusion equations, Arab J. Basic Appl. Sci., 27(1) (2020), 27–44.
[34] M. Meddahi, H. Jafari, and M.N. Ncube, New general integral transform via Atangana–Baleanu derivatives, Adv. Differ. Equ., 2021 (2021).
[35] M. Merdan, On the solutions fractional Riccati differential equation with modified Riemann-Liouville derivative, Int. J. Differ. Equ., 2012 (2012), Article ID 346089, 17 pages.
[36] M. Mohand and A. Mahgoub, The new integral transform Mohand transform, Adv. Theore. Appl. Math., 12(2) (2017), 113–120.
[37] I. Podlubny, Fractional differential equations, San Diego, Academic Press, 1999.
[38] M. S. Rawashdeh, The fractional natural decomposition method: theories and applications, Math. Meth. Appl. Sci., 40 (2016), 2362–237.
[39] N.A. Shah, I. Dassios, E.R. El-Zahar, J.D. Chung, and S. Taherifar, The variational iteration transform method for solving the time-fractional Fornberg–Whitham equation and comparison with decomposition transform method, Mathematics 9(2) (2021), 141.
[40] K. Shah, M. Junaid, and N. Ali, Extraction of Laplace, Sumudu, Fourier and Mellin transform from the natural transform, J. Appl. Envron. Biol. Sci., 5(9) (2015), 1–10.
[41] B. K. Singh and V. K. Srivastava, Approximate series solution of multi-dimensional, time fractional-order (heatlike) diffusion equations using FRDTM., Royal Soc. Open Sci., 2 (2018), 140–511.
[42] G.K. Watugala, Sumudu transform: a new integral transform to solve differential equations and control engineering problems, Int. J. Math. Educat. Sci. Technol., 24(1) (1993), 35–43.
[43] G.C. Wu and D. Baleanu, Variational iteration method for fractional calculus-a universal approach by Laplace transform, Adv. Differ. Equ., 2013 (2013).
[44] D. Ziane, T.M. Elzaki, and M.H. Cherif, Elzaki transform combined with variational iteration method for partial differential equations of fractional order, Fund. J. Math. Appl., 1(1) (2018), 102–108.