Solving class of mixed nonlinear multi-term fractional Volterra-Fredholm integro-differential equations by new development of HAM
Abstract
This work implements the standard Homotopy Analysis Method (HAM) developed by Professor Shijun Liao (1992), and a new development of the HAM (called ND-HAM) improved by Z.K. Eshkuvatov (2022) in solving mixed nonlinear multi-term fractional derivative of different orders of Volterra-Fredholm Integrodifferential equations (FracVF-IDEs). Other than that, the existance and uniqueness of solution as well as the norm convergence with respect to ND-HAM, were proven in a Hilbert space. In addition, three numerical examples (including multi-term fractional IDEs) are presented and compared with the HAM, modified HAM and ”Generalized block pulse operational differentiation matrices method” developed in previous works by illustrating the accuracy as well as validity with respect to ND-HAM. Empirical investigations reveal that ND-HAM and the modified HAM yields the same results when control parameter ℏ is chosen as ℏ = −1 and is comparable to the standard HAM. The findings discovered that the ND-HAM is highly convenient, effective, as well as in line with theoretical results.
Keyword : homotopy analysis method (HAM), new development of HAM (ND-HAM), integro-differential equation (IDEs), Caputo fractional derivative, convergence
How to Cite
Eshkuvatov, Z., Laadjal, Z., & Ismail, S. (2025). Solving class of mixed nonlinear multi-term fractional Volterra-Fredholm integro-differential equations by new development of HAM. Mathematical Modelling and Analysis, 30(1), 52–73. https://doi.org/10.3846/mma.2025.20268
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
H. Akhadkulov, Z. Eshkuvatov, S. Akhatkulov and U.A.M. Roslan. On the approximate solutions of fractional hybrid differential equations. AIP Conference Proceedings, 2746(1):060009, 2023. https://doi.org/10.1063/5.0152396
S. Al-Ahmad, I.M. Sulaiman and M. Mamat. An efficient modification of differential transform method for solving integral and integro-differential equations. Australian Journal of Mathematical Analysis and Applications, 17(2):1–15, 2020.
K. Al-Khaled and F. Allan. Decomposition method for solving nonlinear integroifferential equations. Journal of Applied Mathematics and Computing, 19(1):415–425, 2005. https://doi.org/10.1007/BF02935815
H. Aminikhah, A.H.R. Sheikhani and H. Rezazadeh. Approximate analytical solutions of distributed order fractional Riccati differential equation. Ain Shams Engineering Journal, 9(4):581–588, 2018. https://doi.org/10.1016/j.asej.2016.03.007
A. Avudainayagam and C. Vani. Wavelet-Galerkin method for integrodifferential equations. Applied Numerical Mathematics, 32(3):247–254, 2000. https://doi.org/10.1016/S0168-9274(99)00026-4
A.S. Bataineh, M.S.M. Noorani and I. Hashim. Homotopy analysis method for singular IVPs of Emden-Fowler type. Communications in Nonlinear Science and Numerical Simulation, 14(4):1121–1131, 2009. https://doi.org/10.1016/j.cnsns.2008.02.004
S.H. Behiry and S.I. Mohamed. Solving high-order nonlinear Volterra-Fredholm integro-differential equations by differential transform method. Natural Science, 4(8):581–587, 2012. https://doi.org/0.4236/ns.2012.48077
Y. Cherruault. Convergence of Adomian’s method. Kybernetes, 18(2):31–38, 1989. https://doi.org/10.1108/eb005812
Z. Eshkuvatov, Z. Laadjal and S. Ismail. Numerical treatment of nonlinear mixed Volterra-Fredholm integro-differential equations of fractional order. AIP Conference Proceedings, 2365(1):020006, 2021. https://doi.org/10.1063/5.0057120
Z.K. Eshkuvatov. New development of homotopy analysis method for non-linear integro-differential equations with initial value problems. Mathematical Modeling and Computing, 9(4):842–859, 2022. https://doi.org/10.23939/mmc2022.04.842
Z.K. Eshkuvatov, M.H. Khadijah and B.M. Taib. Modified HPM for high-order linear fractional integro-differential equations of Fredholm-Volterra type. Journal of Physics: Conference Series, 1132(1):012019, 2018. https://doi.org/10.1088/1742-6596/1132/1/012019
M.M. Hosseini. Taylor-successive approximation method for solving nonlinear integral equations. Journal of Advanced Research in Scientific Computing, 1(2):1–13, 2009.
Z. Khan, H.U. Rasheed, M. Ullah, T. Gul and A. Jan. Analytical and numerical solutions of oldroyd 8-constant fluid in double-layer optical fiber coating. Journal of Coatings Technology and Research, 16(1):235–248, 2019. https://doi.org/10.1007/s11998-018-0113-0
R. Kumar, R. Koundal and S. Ali Shehzad. Modified homotopy perturbation approach for the system of fractional partial differential equations: A utility of fractional Wronskian. Mathematical Methods in the Applied Sciences, 45(2):809–826, 2022. https://doi.org/10.1002/mma.7815
S.J. Liao. The proposed homotopy analysis technique for the solution of nonlinear problems. Shanghai Jiao Tong University, Shanghai, 1992. Ph.D. Thesis.
S.J. Liao. Beyond Perturbation: Introduction to the Homotopy Analysis Method. Boca Raton, Chapman and Hall/CRC Press, New York, 2003.
S.J. Liao. Homotopy analysis method in nonlinear differential equations. Springer Berlin, Heidelberg, 2012. https://doi.org/10.1007/978-3-642-25132-0
J.C. Matos, J.A. Matos, M.J. Rodrigues and P.B. Vasconcelos. Approximating the solution of integro-differential problems via the spectral Tau method with filtering. Applied Numerical Mathematics, 149:164–175, 2020. https://doi.org/10.1016/j.apnum.2019.05.025
C. Milici, G. Dr˘ag˘anescu and J.T. Machado. Introduction to fractional differential equations.Springer,Cham,2019. https://doi.org/10.1007/978-3-030-00895-6
S. Noeiaghdam, D. Sidorov, A.-M. Wazwaz, N. Sidorov and V. Sizikov. The numerical validation of the Adomian decomposition method for solving Volterra integral equation with discontinuous kernels using the CESTAC method. Mathematics, 9(260):1–15, 2021. https://doi.org/10.3390/math9030260
S.K. Panda, T. Abdeljawad and K.K. Swamy. New numerical scheme for solving integral equations via fixed point method using distinct (ω − f)-contractions. Alexandria Engineering Journal, 59(4):2015–2026, 2020. https://doi.org/10.1016/j.aej.2019.12.034
K. Parand, A.A. Aghaei, M. Jani and A. Ghodsi. A new approach to the numerical solution of Fredholm integral equations using least squares-support vector regression. Mathematics and Computers in Simulation, 180:114–128, 2021. https://doi.org/10.1016/j.matcom.2020.08.010
I. Podlubny. Fractional Differential Equations. San Diego, Academic Press, 1999.
A. Roohollahi, B. Ghazanfari and S. Akhavan. Numerical solution of the mixed Volterra–Fredholm integro-differential multi-term equations of fractional order. Journal of Computational and Applied Mathematics, 376(112828):1–19, 2020. https://doi.org/10.1016/j.cam.2020.112828
S. Santra and J. Mohapatra. Analysis of the L1 scheme for a time fractional parabolic–elliptic problem involving weak singularity. Mathematical Methods in the Applied Sciences, 44(2):1529–1541, 2020. https://doi.org/10.1002/mma.6850
S. Santra and J. Mohapatra. A novel finite difference technique with error estimate for time fractional partial integro-differential equation of Volterra type. Journal of Computational and Applied Mathematics, 400:113746, 2022. https://doi.org/10.1016/j.cam.2021.113746
K.M. Shadimetov and D.M. Akhmedov. Approximate solution of a singular integral equation using the Sobolev method. Lobachevskii Journal of Mathematics, 43(2):496–505, 2022. https://doi.org/10.1134/S1995080222050249
H. Vu and N.V. Hoa. Hyers-ulam stability of fuzzy fractional Volterra integral equations with the kernel ψ−function via successive approximation method. Fuzzy Sets and Systems, 419(1):67–98, 2021. https://doi.org/10.1016/j.fss.2020.09.009
G. Wang. Symmetry analysis, analytical solutions and conservation laws of a generalized KdV-Burgers-Kuramoto equation and its fractional version. Fractals, 29(04):2150101, 2021. https://doi.org/10.1142/S0218348X21501012
A.-M. Wazwaz. A reliable treatment for mixed Volterra-Fredholm integral equations. Applied Mathematics and Computation, 127(2):405–414, 2002. https://doi.org/10.1016/S0096-3003(01)00020-0
B.J. West, M. Bolognab and P. Grigolini. Physics of Fractal Operators. New York, Springer, 2003. https://doi.org/10.1007/978-0-387-21746-8
S. Al-Ahmad, I.M. Sulaiman and M. Mamat. An efficient modification of differential transform method for solving integral and integro-differential equations. Australian Journal of Mathematical Analysis and Applications, 17(2):1–15, 2020.
K. Al-Khaled and F. Allan. Decomposition method for solving nonlinear integroifferential equations. Journal of Applied Mathematics and Computing, 19(1):415–425, 2005. https://doi.org/10.1007/BF02935815
H. Aminikhah, A.H.R. Sheikhani and H. Rezazadeh. Approximate analytical solutions of distributed order fractional Riccati differential equation. Ain Shams Engineering Journal, 9(4):581–588, 2018. https://doi.org/10.1016/j.asej.2016.03.007
A. Avudainayagam and C. Vani. Wavelet-Galerkin method for integrodifferential equations. Applied Numerical Mathematics, 32(3):247–254, 2000. https://doi.org/10.1016/S0168-9274(99)00026-4
A.S. Bataineh, M.S.M. Noorani and I. Hashim. Homotopy analysis method for singular IVPs of Emden-Fowler type. Communications in Nonlinear Science and Numerical Simulation, 14(4):1121–1131, 2009. https://doi.org/10.1016/j.cnsns.2008.02.004
S.H. Behiry and S.I. Mohamed. Solving high-order nonlinear Volterra-Fredholm integro-differential equations by differential transform method. Natural Science, 4(8):581–587, 2012. https://doi.org/0.4236/ns.2012.48077
Y. Cherruault. Convergence of Adomian’s method. Kybernetes, 18(2):31–38, 1989. https://doi.org/10.1108/eb005812
Z. Eshkuvatov, Z. Laadjal and S. Ismail. Numerical treatment of nonlinear mixed Volterra-Fredholm integro-differential equations of fractional order. AIP Conference Proceedings, 2365(1):020006, 2021. https://doi.org/10.1063/5.0057120
Z.K. Eshkuvatov. New development of homotopy analysis method for non-linear integro-differential equations with initial value problems. Mathematical Modeling and Computing, 9(4):842–859, 2022. https://doi.org/10.23939/mmc2022.04.842
Z.K. Eshkuvatov, M.H. Khadijah and B.M. Taib. Modified HPM for high-order linear fractional integro-differential equations of Fredholm-Volterra type. Journal of Physics: Conference Series, 1132(1):012019, 2018. https://doi.org/10.1088/1742-6596/1132/1/012019
M.M. Hosseini. Taylor-successive approximation method for solving nonlinear integral equations. Journal of Advanced Research in Scientific Computing, 1(2):1–13, 2009.
Z. Khan, H.U. Rasheed, M. Ullah, T. Gul and A. Jan. Analytical and numerical solutions of oldroyd 8-constant fluid in double-layer optical fiber coating. Journal of Coatings Technology and Research, 16(1):235–248, 2019. https://doi.org/10.1007/s11998-018-0113-0
R. Kumar, R. Koundal and S. Ali Shehzad. Modified homotopy perturbation approach for the system of fractional partial differential equations: A utility of fractional Wronskian. Mathematical Methods in the Applied Sciences, 45(2):809–826, 2022. https://doi.org/10.1002/mma.7815
S.J. Liao. The proposed homotopy analysis technique for the solution of nonlinear problems. Shanghai Jiao Tong University, Shanghai, 1992. Ph.D. Thesis.
S.J. Liao. Beyond Perturbation: Introduction to the Homotopy Analysis Method. Boca Raton, Chapman and Hall/CRC Press, New York, 2003.
S.J. Liao. Homotopy analysis method in nonlinear differential equations. Springer Berlin, Heidelberg, 2012. https://doi.org/10.1007/978-3-642-25132-0
J.C. Matos, J.A. Matos, M.J. Rodrigues and P.B. Vasconcelos. Approximating the solution of integro-differential problems via the spectral Tau method with filtering. Applied Numerical Mathematics, 149:164–175, 2020. https://doi.org/10.1016/j.apnum.2019.05.025
C. Milici, G. Dr˘ag˘anescu and J.T. Machado. Introduction to fractional differential equations.Springer,Cham,2019. https://doi.org/10.1007/978-3-030-00895-6
S. Noeiaghdam, D. Sidorov, A.-M. Wazwaz, N. Sidorov and V. Sizikov. The numerical validation of the Adomian decomposition method for solving Volterra integral equation with discontinuous kernels using the CESTAC method. Mathematics, 9(260):1–15, 2021. https://doi.org/10.3390/math9030260
S.K. Panda, T. Abdeljawad and K.K. Swamy. New numerical scheme for solving integral equations via fixed point method using distinct (ω − f)-contractions. Alexandria Engineering Journal, 59(4):2015–2026, 2020. https://doi.org/10.1016/j.aej.2019.12.034
K. Parand, A.A. Aghaei, M. Jani and A. Ghodsi. A new approach to the numerical solution of Fredholm integral equations using least squares-support vector regression. Mathematics and Computers in Simulation, 180:114–128, 2021. https://doi.org/10.1016/j.matcom.2020.08.010
I. Podlubny. Fractional Differential Equations. San Diego, Academic Press, 1999.
A. Roohollahi, B. Ghazanfari and S. Akhavan. Numerical solution of the mixed Volterra–Fredholm integro-differential multi-term equations of fractional order. Journal of Computational and Applied Mathematics, 376(112828):1–19, 2020. https://doi.org/10.1016/j.cam.2020.112828
S. Santra and J. Mohapatra. Analysis of the L1 scheme for a time fractional parabolic–elliptic problem involving weak singularity. Mathematical Methods in the Applied Sciences, 44(2):1529–1541, 2020. https://doi.org/10.1002/mma.6850
S. Santra and J. Mohapatra. A novel finite difference technique with error estimate for time fractional partial integro-differential equation of Volterra type. Journal of Computational and Applied Mathematics, 400:113746, 2022. https://doi.org/10.1016/j.cam.2021.113746
K.M. Shadimetov and D.M. Akhmedov. Approximate solution of a singular integral equation using the Sobolev method. Lobachevskii Journal of Mathematics, 43(2):496–505, 2022. https://doi.org/10.1134/S1995080222050249
H. Vu and N.V. Hoa. Hyers-ulam stability of fuzzy fractional Volterra integral equations with the kernel ψ−function via successive approximation method. Fuzzy Sets and Systems, 419(1):67–98, 2021. https://doi.org/10.1016/j.fss.2020.09.009
G. Wang. Symmetry analysis, analytical solutions and conservation laws of a generalized KdV-Burgers-Kuramoto equation and its fractional version. Fractals, 29(04):2150101, 2021. https://doi.org/10.1142/S0218348X21501012
A.-M. Wazwaz. A reliable treatment for mixed Volterra-Fredholm integral equations. Applied Mathematics and Computation, 127(2):405–414, 2002. https://doi.org/10.1016/S0096-3003(01)00020-0
B.J. West, M. Bolognab and P. Grigolini. Physics of Fractal Operators. New York, Springer, 2003. https://doi.org/10.1007/978-0-387-21746-8