Symmetry-Based Analysis of Volterra Integral Equations via Lie Group Method
Equivalence Groups, Kernel Normal Forms, and Canonical Reductions
Abstract
This study addresses the application of the Lie symmetry method to solve Volterra-type integral equations by transforming them into equivalent differential equations using the Leibniz rule for differentiation under the integral sign. After the transformation, the resulting equations are analyzed using Lie symmetry tools to find closed-form analytical solutions. The study includes a recent literature review (2020–2025) of the most important analytical and symbolic methods in this field, including the Sawi decomposition method, chain methods, and integral transformations.
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References
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[25] X. Zheng, "Approximate inversion for Abel integral operators of variable exponent and applications to fractional Cauchy problems," Fractional Calculus and Applied Analysis, vol. 25, no. 4, pp. 1585-1603, 2022.
[26] A. M. Mahdy, A. S. Nagdy, K. M. Hashem, and D. S. Mohamed, "A computational technique for solving three-dimensional mixed Volterra–Fredholm integral equations," Fractal and Fractional, vol. 7, no. 2, p. 196, 2023.
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[29] M. Bhatti, A. Shahid, I. E. Sarris, and O. Anwar Bég, "Spectral relaxation computation of Maxwell fluid flow from a stretching surface with quadratic convection and non-Fourier heat flux using Lie symmetry transformations," International Journal of Modern Physics B, vol. 37, no. 09, p. 2350082, 2023.
[30] F. Oliveri, "ReLie: a reduce program for Lie group analysis of differential equations," Symmetry, vol. 13, no. 10, p. 1826, 2021.
[31] R. Luong and J. K. Flake, "Measurement invariance testing using confirmatory factor analysis and alignment optimization: A tutorial for transparent analysis planning and reporting," Psychological methods, vol. 28, no. 4, p. 905, 2023.
[32] M. N. Azese, "Rapid variable-step computation of dynamic convolutions and Volterra-type integro-differential equations: RK45 Fehlberg, RK4," Heliyon, vol. 10, no. 13, 2024.
[33] K. S. Nisar, M. Farman, H. Garg, and M. Abdel-Aty, "Soft Computing with Neutrosophic and fractional order frameworks: A state-of-the-Art review," International Journal of Neutrosophic Science (IJNS), vol. 25, no. 3, 2025.
[34] S. Kumar, V. Jadaun, and W.-X. Ma, "Application of the Lie symmetry approach to an extended Jimbo–Miwa equation in (3+ 1) dimensions," The European Physical Journal Plus, vol. 136, no. 8, pp. 1-30, 2021.
[35] G. González Contreras and A. Yakhno, "Symmetries of Systems with the Same Jacobi Multiplier," Symmetry, vol. 15, no. 7, p. 1416, 2023.
[36] J. N. Fuhg et al., "A review on data-driven constitutive laws for solids," Archives of Computational Methods in Engineering, pp. 1-43, 2024.
[37] A. Georgieva and A. Pavlova, "Fuzzy Sawi decomposition method for solving nonlinear partial fuzzy differential equations," Symmetry, vol. 13, no. 9, p. 1580, 2021.
[38] J. Sochacki and A. Tongen, Applying Power Series to Differential Equations. Springer, 2023.
[39] B. J. Walker, A. K. Townsend, A. K. Chudasama, and A. L. Krause, "VisualPDE: rapid interactive simulations of partial differential equations," Bulletin of Mathematical Biology, vol. 85, no. 11, p. 113, 2023.
[40] S. Kishimoto, N. Ishikawa, and S. Ohnuki, "Analysis of instantaneous acoustic fields using fast Inverse Laplace transform," IEICE Transactions on Electronics, vol. 105, no. 11, pp. 700-703, 2022.
[41] V. Pavlenko, S. Pavlenko, and V. Speranskyy, "Identification of systems using Volterra model in time and frequency domain," in Advanced Data Acquisition and Intelligent Data Processing: River Publishers, 2022, pp. 233-269.
[42] N. Ali, "Algorithm for Visualization of Zero Divisor Graphs of the Ring ℤn Using MAPLE Coding," Open Journal of Discrete Mathematics, vol. 14, no. 1, pp. 1-8, 2024.
[43] E. Kulikov and A. Makarov, "A method for solving the Fredholm integral equation of the first kind," Journal of Mathematical Sciences, vol. 272, no. 4, pp. 558-565, 2023.
[2] S.-P. Hong, "Different numerical techniques, modeling and simulation in solving complex problems," Journal of Machine and Computing, vol. 3, no. 2, pp. 058-086, 2023.
[3] S. Peddinti and Y. Sabbani, "Mathematical Modeling of Infectious Disease Spread Using Differential Equations and Epidemiological Insights," Journal of Nonlinear Analysis and Optimization, vol. 15, pp. 114-122, 2024.
[4] M. Ghasemi, A. Goligerdian, and S. Moradi, "A novel super-convergent numerical method for solving nonlinear Volterra integral equations based on B-splines," Mediterranean Journal of Mathematics, vol. 21, no. 4, p. 129, 2024.
[5] H. HamaRashid, H. M. Srivastava, M. Hama, P. O. Mohammed, E. Al-Sarairah, and M. Y. Almusawa, "New numerical results on existence of Volterra–Fredholm integral equation of nonlinear boundary integro-differential type," Symmetry, vol. 15, no. 6, p. 1144, 2023.
[6] A. A. Harbi, Z. N. Kazem, and S. E. Mohammed, "Advancements in Numerical Analysis: Techniques for Solving Volterra and Fredholm Equations," Journal of Al-Qadisiyah for Computer Science and Mathematics, vol. 17, no. 2, pp. 80–91-80–91, 2025.
[7] R. Saneifard, A. Jafarian, N. Ghalami, and S. M. Nia, "Extended artificial neural networks approach for solving two-dimensional fractional-order Volterra-type integro-differential equations," Information Sciences, vol. 612, pp. 887-897, 2022.
[8] N. H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, Volume I: Symmetries, Exact Solutions, and Conservation Laws. CRC press, 2023.
[9] F. Alizadeh, "Lie Symmetries, Invariant Solutions, and Conservation Laws of the Logarithmic Modified Nonlinear Schrödinger Equation," International Journal of Applied and Computational Mathematics, vol. 11, no. 4, p. 124, 2025.
[10] A. I. Aliyu et al., "Lie Symmetry Analysis and Explicit Solutions to the Estevez–Mansfield–Clarkson Equation," Symmetry, vol. 16, no. 9, p. 1194, 2024.
[11] H. Tang, K. Chong, and B. Kee, "Lie symmetries, optimal system, and invariant solutions of the generalized Cox-Ingersoll-Ross equation," in International Conference on Mathematical Sciences and Statistics 2022 (ICMSS 2022), 2022: Atlantis Press, pp. 103-113.
[12] M. Usman, A. Hussain, F. Zaman, and N. Abbas, "Symmetry analysis and invariant solutions of generalized coupled Zakharov-Kuznetsov equations using optimal system of Lie subalgebra," Int. J. Math. Comput. Eng., vol. 2, no. 2, pp. 53-70, 2024.
[13] W. A. Faridi and S. A. AlQahtani, "The explicit power series solution formation and computationof Lie point infinitesimals generators: Lie symmetry approach," Physica Scripta, vol. 98, no. 12, p. 125249, 2023.
[14] V. Ibrahimov, M. Imanova, and D. Juraev, "Application of the bilateral hybrid methods to solving initial-value problems for the Volterra integro-differential equations," WSEAS Transactions on Mathematics, vol. 22, pp. 781-791, 2023.
[15] M. Vaquero, J. Cortés, and D. M. de Diego, "Symmetry preservation in Hamiltonian systems: simulation and learning," Journal of Nonlinear Science, vol. 34, no. 6, p. 115, 2024.
[16] S. D. Koval, A. Bihlo, and R. O. Popovych, "Extended symmetry analysis of remarkable (1+ 2)-dimensional Fokker–Planck equation," European Journal of Applied Mathematics, vol. 34, no. 5, pp. 1067-1098, 2023.
[17] H. Hilmi, S. J. MohammedFaeq, and S. S. Fatah, "Exact and approximate solution of multi-higher order fractional differential equations via sawi transform and sequential approximation method," Journal of University of Babylon for Pure and Applied Sciences, pp. 311-334, 2024.
[18] A.-K. Alomari, M. Alaroud, N. Tahat, and A. Almalki, "Extended Laplace power series method for solving nonlinear Caputo fractional Volterra integro-differential equations," Symmetry, vol. 15, no. 7, p. 1296, 2023.
[19] M. A. Hussein, H. K. Jassim, and A. K. Jassim, "An innovative iterative approach to solving Volterra integral equations of second kind," ACTA Polytechnica, vol. 64, no. 2, 2024.
[20] D. Stocco, "Symbolic Computation Methods for the Numerical Solution of Dynamic Systems Described by Differential-Algebraic Equations," 2024.
[21] B. Ghanbari, S. Kumar, M. Niwas, and D. Baleanu, "The Lie symmetry analysis and exact Jacobi elliptic solutions for the Kawahara–KdV type equations," Results in Physics, vol. 23, p. 104006, 2021.
[22] D. H. S. Perera and D. Gallage, "Solution Methods for Nonlinear Ordinary Differential Equations Using Lie Symmetry Groups," Advanced Journal of Graduate Research, vol. 13, no. 1, pp. 37-61, 2023.
[23] K. Padmaja, B. R. Kumar, A. V. Kumar, and R. S. Gorla, "A comprehensive review of mathematical methods for fluid flow, heat and mass transfer problems: pros, cons and key findings: K. Padmaja et al," Journal of Thermal Analysis and Calorimetry, pp. 1-27, 2025.
[24] 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, vol. 9, no. 3, p. 260, 2021.
[25] X. Zheng, "Approximate inversion for Abel integral operators of variable exponent and applications to fractional Cauchy problems," Fractional Calculus and Applied Analysis, vol. 25, no. 4, pp. 1585-1603, 2022.
[26] A. M. Mahdy, A. S. Nagdy, K. M. Hashem, and D. S. Mohamed, "A computational technique for solving three-dimensional mixed Volterra–Fredholm integral equations," Fractal and Fractional, vol. 7, no. 2, p. 196, 2023.
[27] B. Mordukhovich and P. Pérez-Aros, "Generalized Leibniz rules and Lipschitzian stability for expected-integral mappings," SIAM Journal on Optimization, vol. 31, no. 4, pp. 3212-3246, 2021.
[28] R. T. Arthur and D. Rabouin, Leibniz on the Foundations of the Differential Calculus. Springer, 2025.
[29] M. Bhatti, A. Shahid, I. E. Sarris, and O. Anwar Bég, "Spectral relaxation computation of Maxwell fluid flow from a stretching surface with quadratic convection and non-Fourier heat flux using Lie symmetry transformations," International Journal of Modern Physics B, vol. 37, no. 09, p. 2350082, 2023.
[30] F. Oliveri, "ReLie: a reduce program for Lie group analysis of differential equations," Symmetry, vol. 13, no. 10, p. 1826, 2021.
[31] R. Luong and J. K. Flake, "Measurement invariance testing using confirmatory factor analysis and alignment optimization: A tutorial for transparent analysis planning and reporting," Psychological methods, vol. 28, no. 4, p. 905, 2023.
[32] M. N. Azese, "Rapid variable-step computation of dynamic convolutions and Volterra-type integro-differential equations: RK45 Fehlberg, RK4," Heliyon, vol. 10, no. 13, 2024.
[33] K. S. Nisar, M. Farman, H. Garg, and M. Abdel-Aty, "Soft Computing with Neutrosophic and fractional order frameworks: A state-of-the-Art review," International Journal of Neutrosophic Science (IJNS), vol. 25, no. 3, 2025.
[34] S. Kumar, V. Jadaun, and W.-X. Ma, "Application of the Lie symmetry approach to an extended Jimbo–Miwa equation in (3+ 1) dimensions," The European Physical Journal Plus, vol. 136, no. 8, pp. 1-30, 2021.
[35] G. González Contreras and A. Yakhno, "Symmetries of Systems with the Same Jacobi Multiplier," Symmetry, vol. 15, no. 7, p. 1416, 2023.
[36] J. N. Fuhg et al., "A review on data-driven constitutive laws for solids," Archives of Computational Methods in Engineering, pp. 1-43, 2024.
[37] A. Georgieva and A. Pavlova, "Fuzzy Sawi decomposition method for solving nonlinear partial fuzzy differential equations," Symmetry, vol. 13, no. 9, p. 1580, 2021.
[38] J. Sochacki and A. Tongen, Applying Power Series to Differential Equations. Springer, 2023.
[39] B. J. Walker, A. K. Townsend, A. K. Chudasama, and A. L. Krause, "VisualPDE: rapid interactive simulations of partial differential equations," Bulletin of Mathematical Biology, vol. 85, no. 11, p. 113, 2023.
[40] S. Kishimoto, N. Ishikawa, and S. Ohnuki, "Analysis of instantaneous acoustic fields using fast Inverse Laplace transform," IEICE Transactions on Electronics, vol. 105, no. 11, pp. 700-703, 2022.
[41] V. Pavlenko, S. Pavlenko, and V. Speranskyy, "Identification of systems using Volterra model in time and frequency domain," in Advanced Data Acquisition and Intelligent Data Processing: River Publishers, 2022, pp. 233-269.
[42] N. Ali, "Algorithm for Visualization of Zero Divisor Graphs of the Ring ℤn Using MAPLE Coding," Open Journal of Discrete Mathematics, vol. 14, no. 1, pp. 1-8, 2024.
[43] E. Kulikov and A. Makarov, "A method for solving the Fredholm integral equation of the first kind," Journal of Mathematical Sciences, vol. 272, no. 4, pp. 558-565, 2023.
Published
2026-02-04
Section
Research Articles
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