# Vectorial Iterative Fractional Laplace Transform Method for the Analytic Solutions of Fractional Cauchy-Riemann Systems Partial Differential Equations

## Main Article Content

## Abstract

The present study aims to obtain infinite fractional power series solution vectors of fractional Cauchy-Riemann systems equations with initial conditions by the use of vectorial iterative fractional Laplace transform method (VIFLTM). The basic idea of the VIFLTM was developed successfully and applied to four test examples to see its effectiveness and applicability. The infinite fractional power series form solutions were successfully obtained analytically. Thus, the results show that the VIFLTM works successfully in solving fractional Cauchy-Riemann system equations with initial conditions, and hence it can be extended to other fractional differential equations.

Keywords:

Fractional Cauchy-Riemann systems equations, Caputo fractional derivatives, vectorial iterative fractional Laplace transform method.

## Article Details

How to Cite

*Asian Research Journal of Mathematics*,

*16*(1), 60-83. https://doi.org/10.9734/arjom/2020/v16i130169

Section

Original Research Article

## References

El-Ajou A, Arqub OA, Al-Zhour Z, Momani S. New results on fractional power series theories and applications. Entropy. 2013;15(12):5305-5323.

Millar K, Ross B. An introduction to the fractional calculus and fractional differential equations. New York, USA: John Wiley and Sons; 1993.

Oldham KB, Spanier J. The fractional calculus: Theory and applications of differentiation and integration of arbitrary order. R. Bellman, Ed., New York: Academic Press. 1974;111.

Podlubny I. Fractional differential equations. W. Y. Ames, Ed., San Diego, London: Academic Press. 1999;198.

Diethelm K. The analysis of fractional differential equations., J. M. Cachan, F. T. Groningen and B. T. Paris, Eds., Verlag, Berlin, Heidelberg: Springer. 2010;2004.

Hilfer R, Ed. Applications of fractional calculus in physics. Singapore, New Jersey, London, Hong Kong: World Scientific; 2000.

Lazarevic M, Rapaic MR, Sekara TB, Stojanovic SB, Lj Debeljkovic D, Vosika Z, Lazovic G, Simic-Krstic J, Koruga D, Spasic DT, Hedrih AN, Stevanovic Hedrih KR. Advanced topics on applications of fractional calculus on control problems, system stability and modeling. Belgrade: WSEAS Press; 2014.

Kumar M, Saxena SA. Recent advancement in fractional calculus. Advance Technology in Engineering and Science. 2016;4(4):177-186.

Ross B, Ed. Fractional calculus and its applications. Berlin, Heidelberg, New York: Springer-Verlag. 1975;457.

Mainardi F. Fractional calculus and waves in linear viscoelasticity. An introduction to mathematical models. London: Imperial College Press; 2010.

Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. J. V. Mil, Ed., Amsterdam: Elsevier. 2006;204.

Das S. Functional fractional calculus, 2nd Ed. Velag, Berlin, Heidelberd: Springer; 2011.

David S, Linarese J, Pallone E. Fractional order calculus: historical apologia: Basic concepts and some applications. Revista Brasileira de Ensino de Fisica. 2011;33(4):4302-7.

Sabatier J, Agrawal O, Machado J, Eds. Advances in fractional calculus. Theoretical developments and applications in physics and engineering. Dordrecht: Springer; 2007.

Sheng H, Chen Y, Qiu T. Fractional processes and fractional-order signal processing; Techniques and applications. Verlag: Springer; 2011.

Rahimy M. Applications of fractional differential equations. Applied Mathematics Sciences. 2010;4(50):2453-2461.

David SA, Katayama AH. Fractional order ForFood gums: Modeling and simulation. Applied Mathematics. 2013;4:305-309.

Tarsov VE. Fractional dynamics: Application of fractional calculus to dynamics of particles, fields and media. A. C. Luo and N. H. Ibragimov, Eds., Heidelberg, Dordrecht, London, New York: Springer; 2011.

Zaslavsky GM. Himaltonian chaos and fractional dynamics. New York: Oxford University Press; 2005.

Magin R. Modeling the cardiac tissue electrode interface using fractional calculus. Journal of Vibration and Control. 2008;14(9-10):1431–1442.

Margin RL. Fractional culculus models of complex dynamics in biological tissues. Computers and Mathematics with Applications. 2010;59:1586-1593.

Dalir M, Bashour M. Applications of fractional calculus. Applied Mathematical Sciences. 2010;4(22): 1021-1032.

Gomez-Aguilar JF, Yepez-Martinez H, Calderon-Ramon C, Cruz-Orduna I, Escobar-Jimenez RF, Olivares-Peregrino VH. Modeling of a mass-spring-damper system by fractional derivatives with and without a singular Kernel. Entropy. 2015;17:6289-6303.

Das AK, Roy TK. Role of fractional calculus to the generalized inventory model. Journal of Global Research in Computer Science. 2014;5(2):11-23.

Carpinteri A, Cornetti P, Sapora A. Nonlocality: An approach based on fractional calculus. Meccanica. 2014;49:2551-2569.

Duan JS. A modifed fractional derivative and its application to fractional vibration. Appl. Math. Inf. Sci. 2016;10(5):1863-1869.

Koeller R. Applications of fractional calculus to the theory of viscoelasticity. J. Appl. Mech. 1984;51:299-307.

Metzler R, Klafter J. The random walk's guide to anomalous diffusion: A fractional dynamics approach. Elsevier, Amsterdam; 2000.

Da Silva L, Tateishi A, Lenzi M, Lenzi E, da Silva P. Green function for a non-Markovian Fokker-Planck equation: Comb-model and anomalous diffusion. Brazilian Journal of Physics. 2009;39(2A):483-487.

Shah NET, Animasaun I, MY. Insight into the natural convection flow through a vertical cylinder using caputo time-fractional derivatives. Int. J. Appl. Comput. Math. 2018;4(80).

Shah NA, Hajizadeh A, Zeb M, Sohail A, Mahsud Y, Animasaun IL. Effect of magnetic field on double convection flow of viscous fluid over a moving vertical plate with constant temperature and general concentration by using new trend of fractional derivative. Open J. Math. Sci. 2018;2:253-265.

El-Ajou A, Abu Arqub O, Al-S M. A general form of the generalized Taylor’s formula with some applications. Applied Mathematics and Computation. 2015;256:851-859.

Gorenflo R, Luchko Y, Yamamoto M. Time-fractional diffusion equation in the fractional Sbololev spaces. Fractional Calculus & Applied Analysis. 2015;18(3):799–820.

Luchko L. A new fractional calculus model for the two-dimensional anomalous diffusion and its analysis. Math. Model. Nat. Phenom. 2016;11(3):1-17.

Zeid SS, Yousefi M, Kamyad AV. Approximate solutions for a class of fractional order model of HIV infection via Linear programming problem. American Journal of Computational Mathematics. 2016;6:141-152.

Abu Arqub O, El-Ajou A, Momani S. Constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations. Journal of Computational Physics. 2015;293:385-399.

Kumar A, Kumar S, Yan SP. Residual power series method for fractional diffusion equations. Fundamenta Informaticae. 2017;151:213-230.

Cetinkaya A, Kiymaz O. The solution of the time-fractional diffusion equation by the generalized differential transform method. Mathematical and Computer Modelling. 2013;57:2349–2354.

Kumar S, Yildirim A, Khan Y, Wei L. A fractional model of the diffusion equation and its analytical solution using Laplace transform. Scientia Iranica B. 2012;19(4):1117–1123.

Kebede SK. Analytic solutions of time fractional diffusion equations by fractional reduced differential transform method. African Journal of Mathematics and Computer Science Research. 2018;11(2):14-34.

Kebede SK. Analytic solutions of (N+1) dimensional time fractional diffusion equations by Iterative fractional Laplace transform method. Global Journals of Science Frontier Research: Mathematics and Decision Sciences. 2018;4:29-54.

Wang Y. Generalized viscoelastic wave equation. Geophysical Journal International. 2016;204:1216–1221.

Abu Arqub O. Fitted reproducing kernel Hilbert space method for the solutions of some certain classes of time-fractional partial differential equations subject to initial and Neumann boundary conditions. Computers & Mathematics with Applications. 2017;73:1243-1261.

AMS, Mahdy MGMA. Fractional complex transform for solving the fractional differential equations. Global Journal of Pure and Applied Mathematics. 2018;14(1):17-37.

Korbel J, Luchko Y. Modeling of financial processes with a space-time fractional diffusion equation of varying order. An International Journal for Theory and Applications. 2016;19(6):1414-1433.

Iyiola OS, Zaman FD. A fractional diffusion equation model for cancer tumor. AIP Advances. 2014;4:107121-4.

Wendland W. Elliptic systems in the plane. London: Pitman; 1979.

Naseem T, Tahir M. Vectorial reduced differential transform (VRDT) method for the solution of inhomogeneous Cauchy-Riemann system. International Journal of Physical Sciences. 2014;9(2):20-25.

Hadamard J. Lectures on Cauchy’s problem in linear partial differential equations. New Haven: Yale University Press; 1923.

Farmer C, Howison S. The motion of a viscous filament in a Hele–Shaw cell: A physical realisation of the Cauchy–Riemann equations. Applied Mathematics Letters. 2006;19:356–361.

Joseph DD, Saut JC. Short-wave instabilities and Ill-posed initial-value problems. Theoret. Comput. Fluid Dynamics. 1990;1:191-227.

Reichel L. Numerical methods for analytic continuation and mesh generation. Constr. Approx. 1986;2:23-39.

Gustafson S. Convergence acceleration on a general class of power series. Computing. 1978;21:53-69.

Henrici P. An algorithm for analytic continuation. SIAM J. Numer. Anal. 1966;3:67-78.

Caputo M, Mainardi F. Linear models of dissipation in anelastic solids. Rivista del Nuovo Cimento. 1971;1(2):161-198.

Caputo M. Linear models of dissipation whose Q is almost frequency independent: Part II. Geophys. J. R. Ustr. SOC. 1967;13:529-539.

Daftardar-Gejji V, Jafari H. An iterative method for solving nonlinear functional equations. J. Math. Anal. Appl. 2006;316:753-763.

Daftardar-Gejji V, Bhalekar S. Solving fractional boundary value problems with Dirichlet boundary conditions using a new iterative method. Computers and Mathematics with Applications. 2010;59:1801-1809.

Jafari H, Nazari M, Baleanu D, Khalique C. A new approach for solving a system of fractional partial differential equations. Computers and Mathematics with Applications. 2013;66:838–843.

Yan L. Numerical solutions of fractional Fokker-Planck equations using iterative Laplace transform method. Abstract and Applied Analysis. 2013;7.

Sharma S, Bairwa R. Iterative Laplace transform method for solving fractional heat and wave-like equations. Research Journal of Mathematical and Statistical Sciences. 2015;3(2):4-9.

Sharma SC, Bairwa RK. Exact solution of generalized time-fractional biological population model by means of iterative Laplace transform method. International Journal of Mathematical Archive. 2014;5(12):40-46.

Millar K, Ross B. An introduction to the fractional calculus and fractional differential equations. New York, USA: John Wiley and Sons; 1993.

Oldham KB, Spanier J. The fractional calculus: Theory and applications of differentiation and integration of arbitrary order. R. Bellman, Ed., New York: Academic Press. 1974;111.

Podlubny I. Fractional differential equations. W. Y. Ames, Ed., San Diego, London: Academic Press. 1999;198.

Diethelm K. The analysis of fractional differential equations., J. M. Cachan, F. T. Groningen and B. T. Paris, Eds., Verlag, Berlin, Heidelberg: Springer. 2010;2004.

Hilfer R, Ed. Applications of fractional calculus in physics. Singapore, New Jersey, London, Hong Kong: World Scientific; 2000.

Lazarevic M, Rapaic MR, Sekara TB, Stojanovic SB, Lj Debeljkovic D, Vosika Z, Lazovic G, Simic-Krstic J, Koruga D, Spasic DT, Hedrih AN, Stevanovic Hedrih KR. Advanced topics on applications of fractional calculus on control problems, system stability and modeling. Belgrade: WSEAS Press; 2014.

Kumar M, Saxena SA. Recent advancement in fractional calculus. Advance Technology in Engineering and Science. 2016;4(4):177-186.

Ross B, Ed. Fractional calculus and its applications. Berlin, Heidelberg, New York: Springer-Verlag. 1975;457.

Mainardi F. Fractional calculus and waves in linear viscoelasticity. An introduction to mathematical models. London: Imperial College Press; 2010.

Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. J. V. Mil, Ed., Amsterdam: Elsevier. 2006;204.

Das S. Functional fractional calculus, 2nd Ed. Velag, Berlin, Heidelberd: Springer; 2011.

David S, Linarese J, Pallone E. Fractional order calculus: historical apologia: Basic concepts and some applications. Revista Brasileira de Ensino de Fisica. 2011;33(4):4302-7.

Sabatier J, Agrawal O, Machado J, Eds. Advances in fractional calculus. Theoretical developments and applications in physics and engineering. Dordrecht: Springer; 2007.

Sheng H, Chen Y, Qiu T. Fractional processes and fractional-order signal processing; Techniques and applications. Verlag: Springer; 2011.

Rahimy M. Applications of fractional differential equations. Applied Mathematics Sciences. 2010;4(50):2453-2461.

David SA, Katayama AH. Fractional order ForFood gums: Modeling and simulation. Applied Mathematics. 2013;4:305-309.

Tarsov VE. Fractional dynamics: Application of fractional calculus to dynamics of particles, fields and media. A. C. Luo and N. H. Ibragimov, Eds., Heidelberg, Dordrecht, London, New York: Springer; 2011.

Zaslavsky GM. Himaltonian chaos and fractional dynamics. New York: Oxford University Press; 2005.

Magin R. Modeling the cardiac tissue electrode interface using fractional calculus. Journal of Vibration and Control. 2008;14(9-10):1431–1442.

Margin RL. Fractional culculus models of complex dynamics in biological tissues. Computers and Mathematics with Applications. 2010;59:1586-1593.

Dalir M, Bashour M. Applications of fractional calculus. Applied Mathematical Sciences. 2010;4(22): 1021-1032.

Gomez-Aguilar JF, Yepez-Martinez H, Calderon-Ramon C, Cruz-Orduna I, Escobar-Jimenez RF, Olivares-Peregrino VH. Modeling of a mass-spring-damper system by fractional derivatives with and without a singular Kernel. Entropy. 2015;17:6289-6303.

Das AK, Roy TK. Role of fractional calculus to the generalized inventory model. Journal of Global Research in Computer Science. 2014;5(2):11-23.

Carpinteri A, Cornetti P, Sapora A. Nonlocality: An approach based on fractional calculus. Meccanica. 2014;49:2551-2569.

Duan JS. A modifed fractional derivative and its application to fractional vibration. Appl. Math. Inf. Sci. 2016;10(5):1863-1869.

Koeller R. Applications of fractional calculus to the theory of viscoelasticity. J. Appl. Mech. 1984;51:299-307.

Metzler R, Klafter J. The random walk's guide to anomalous diffusion: A fractional dynamics approach. Elsevier, Amsterdam; 2000.

Da Silva L, Tateishi A, Lenzi M, Lenzi E, da Silva P. Green function for a non-Markovian Fokker-Planck equation: Comb-model and anomalous diffusion. Brazilian Journal of Physics. 2009;39(2A):483-487.

Shah NET, Animasaun I, MY. Insight into the natural convection flow through a vertical cylinder using caputo time-fractional derivatives. Int. J. Appl. Comput. Math. 2018;4(80).

Shah NA, Hajizadeh A, Zeb M, Sohail A, Mahsud Y, Animasaun IL. Effect of magnetic field on double convection flow of viscous fluid over a moving vertical plate with constant temperature and general concentration by using new trend of fractional derivative. Open J. Math. Sci. 2018;2:253-265.

El-Ajou A, Abu Arqub O, Al-S M. A general form of the generalized Taylor’s formula with some applications. Applied Mathematics and Computation. 2015;256:851-859.

Gorenflo R, Luchko Y, Yamamoto M. Time-fractional diffusion equation in the fractional Sbololev spaces. Fractional Calculus & Applied Analysis. 2015;18(3):799–820.

Luchko L. A new fractional calculus model for the two-dimensional anomalous diffusion and its analysis. Math. Model. Nat. Phenom. 2016;11(3):1-17.

Zeid SS, Yousefi M, Kamyad AV. Approximate solutions for a class of fractional order model of HIV infection via Linear programming problem. American Journal of Computational Mathematics. 2016;6:141-152.

Abu Arqub O, El-Ajou A, Momani S. Constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations. Journal of Computational Physics. 2015;293:385-399.

Kumar A, Kumar S, Yan SP. Residual power series method for fractional diffusion equations. Fundamenta Informaticae. 2017;151:213-230.

Cetinkaya A, Kiymaz O. The solution of the time-fractional diffusion equation by the generalized differential transform method. Mathematical and Computer Modelling. 2013;57:2349–2354.

Kumar S, Yildirim A, Khan Y, Wei L. A fractional model of the diffusion equation and its analytical solution using Laplace transform. Scientia Iranica B. 2012;19(4):1117–1123.

Kebede SK. Analytic solutions of time fractional diffusion equations by fractional reduced differential transform method. African Journal of Mathematics and Computer Science Research. 2018;11(2):14-34.

Kebede SK. Analytic solutions of (N+1) dimensional time fractional diffusion equations by Iterative fractional Laplace transform method. Global Journals of Science Frontier Research: Mathematics and Decision Sciences. 2018;4:29-54.

Wang Y. Generalized viscoelastic wave equation. Geophysical Journal International. 2016;204:1216–1221.

Abu Arqub O. Fitted reproducing kernel Hilbert space method for the solutions of some certain classes of time-fractional partial differential equations subject to initial and Neumann boundary conditions. Computers & Mathematics with Applications. 2017;73:1243-1261.

AMS, Mahdy MGMA. Fractional complex transform for solving the fractional differential equations. Global Journal of Pure and Applied Mathematics. 2018;14(1):17-37.

Korbel J, Luchko Y. Modeling of financial processes with a space-time fractional diffusion equation of varying order. An International Journal for Theory and Applications. 2016;19(6):1414-1433.

Iyiola OS, Zaman FD. A fractional diffusion equation model for cancer tumor. AIP Advances. 2014;4:107121-4.

Wendland W. Elliptic systems in the plane. London: Pitman; 1979.

Naseem T, Tahir M. Vectorial reduced differential transform (VRDT) method for the solution of inhomogeneous Cauchy-Riemann system. International Journal of Physical Sciences. 2014;9(2):20-25.

Hadamard J. Lectures on Cauchy’s problem in linear partial differential equations. New Haven: Yale University Press; 1923.

Farmer C, Howison S. The motion of a viscous filament in a Hele–Shaw cell: A physical realisation of the Cauchy–Riemann equations. Applied Mathematics Letters. 2006;19:356–361.

Joseph DD, Saut JC. Short-wave instabilities and Ill-posed initial-value problems. Theoret. Comput. Fluid Dynamics. 1990;1:191-227.

Reichel L. Numerical methods for analytic continuation and mesh generation. Constr. Approx. 1986;2:23-39.

Gustafson S. Convergence acceleration on a general class of power series. Computing. 1978;21:53-69.

Henrici P. An algorithm for analytic continuation. SIAM J. Numer. Anal. 1966;3:67-78.

Caputo M, Mainardi F. Linear models of dissipation in anelastic solids. Rivista del Nuovo Cimento. 1971;1(2):161-198.

Caputo M. Linear models of dissipation whose Q is almost frequency independent: Part II. Geophys. J. R. Ustr. SOC. 1967;13:529-539.

Daftardar-Gejji V, Jafari H. An iterative method for solving nonlinear functional equations. J. Math. Anal. Appl. 2006;316:753-763.

Daftardar-Gejji V, Bhalekar S. Solving fractional boundary value problems with Dirichlet boundary conditions using a new iterative method. Computers and Mathematics with Applications. 2010;59:1801-1809.

Jafari H, Nazari M, Baleanu D, Khalique C. A new approach for solving a system of fractional partial differential equations. Computers and Mathematics with Applications. 2013;66:838–843.

Yan L. Numerical solutions of fractional Fokker-Planck equations using iterative Laplace transform method. Abstract and Applied Analysis. 2013;7.

Sharma S, Bairwa R. Iterative Laplace transform method for solving fractional heat and wave-like equations. Research Journal of Mathematical and Statistical Sciences. 2015;3(2):4-9.

Sharma SC, Bairwa RK. Exact solution of generalized time-fractional biological population model by means of iterative Laplace transform method. International Journal of Mathematical Archive. 2014;5(12):40-46.