Approximating Higher Order Linear Fredholm Integro-Differential Equations by an Efficient Adomian Decomposition Method

Year 2024, Volume: 2 Issue: 1, 44 - 54, 25.06.2024

Kabiru Oyeleye Kareem , Morufu Oyedunsi Olayiwola , Muideen Odunayo Ogunniran , Asimiyu Olalekan Oladapo , Akeem Olanrewaju Yunus , Kamilu Adewale Adedokun , Joseph Adeleke Adedeji , Ismail Adedapo Alaje

https://doi.org/10.26650/ijmath.2024.00015

Abstract

This work presents a unique technique for the precise and efficient solution of Linear Fredholm integro-differential equations (LFDEs), the technique is based on the Modification of Adomian Decomposition Method (MADM). The MADM extends the well-known Adomian Decomposition Method (ADM) by integrating novel changes that improve convergence and computing efficiency. The LFDEs are essential for simulating a wide range of phenomena in science and engineering. Because their analytical solutions are frequently difficult to achieve, the development of efficient and trustworthy numerical approaches is required. We present an introduction of the MADM method and its important characteristics emphasizing its capacity to handle a wide range of LFDEs seen in scientific and engineering applications. We demonstrate the method’s usefulness in contrast to the true approach, stressing its computational benefits and precision.

Keywords

Fredholm Integro, differential Equations, Numerical Solutions, Computational Efficiency

References

• Abdella, K., Ross G., 2020, Solving integro-differential boundary value problems using sinc-derivative collocation, MDPI, 8(1637): 1-13. google scholar
• Acar, N. I., and Dascioglu, A. 2019, A projection method for linear Fredolm-Volterra integro-differential equations, J. Taibah Univ. Sci., 13, 644-650. google scholar
• Akyuz A., 2006, Chebyshev polynomial approach for linear Fredholm-Volterra integro-differential equations in the most general form, Applied Mathematics Computation, 181(1): 103-112. doi: 10.1016/j.amc.2006.01.018. google scholar
• Amin, R., Shah, K., Asif, M. and Khan, I., 2020, Efficient numerical technique for solution of delay Volterra-Fredholm integral equations using Haar wavelet, Heliyon 6, e05108. google scholar
• Ayinde, A. M., James A. A., Ishaq A. A. and Oyedepo T., 2022, A new numerical approach using Chebyshev third kind polynomial for solving integro-differential equations of higher order, Gazi University Journal of Sciences, Part A, 9(3): 259-266. google scholar
• Bogdan Caruntu and Madalina Sofia Pasca 2021, Approximation Solution for a class of Nonlinear Fredholm and Volterra Integro-Differential Equations using the Polynomial Least Squares Method, Mathematic 9, 2692, 1-13. google scholar
• Buranay, S. C., Ozarslan, M. A. and Falahesar, S. S. 2021, Numerical Solution of the Fredholm and Volterra integral equations by using modified Berastein-Kantonowich operators, Mathematics, 9, 1193. google scholar
• Davaeeifar, S. and Rashidainia J., 2017, Boubakar polynomials collocation approach for solving systems of nonlinear Volterra-Fredholm integral equations, J. Taibah Uni. Sci., 11, 1182-1199. google scholar
• El-Hawary, H. M. and El-Sheshtawy, T. S. 2010, Spectral method for solving the general form linear Fredholm Volterra integro-differential equations based on Chebyshev polynomials, J. Mod. Met. Numer. Math., 1, 1-11. google scholar
• Hosry, A., Nakad, R. and Bhalekar, S., 2020, A hybrid fuction approach to solving a class of Fredholm and Volterra integro-differential equations, Math. Comput. Appl., 25, 1-16. google scholar
• Kabiru Kareem, Morufu Olayiwola, Oladapo Asimiyu, Yunus Akeem, Kamilu Adedokun and Ismail Alaje, 2023, On the solution of volterra integro-differential Equations using a Modified Adomian Decomposition Method, Jambura Journal of Mathematics, 5(2): 265-277. google scholar
• Kabiru Oyeleye Kareem, Oyedunsi Olayiwola and Muideen Odunayo Ogunniran, 2023, On the numerical solution of Fredholm-type integro-differential equations using an efficient modified Adomian decomposition method, Mathematics and Computational Sciences, 4(4): 39-52. google scholar
• Kamoh N. M., Gyemang, D. G. and Soomiyol, M. C., 2019, Comparing the efficiency of Simpson’s 1/3 and Simpson’s 3/8 rules for the numerical solution of first order Volterra Integro-differential equations, World Academy of Science, Engineering and Technology International, Journal of Mathematical and Computational Sciences, 13(5): 136-139. google scholar
• Kurkou, O. K., Aslan E. and Sezer M., 2017, A novel collocation method based on residual error analysis for solving integro-differential equations using hybrid Dickson and Taylor polynomials, Sains Malaysiana, 46, 335-347. google scholar
• Lofti, M. and Alipanah A., 2020, “Legendre spectral element method for solving Volterra integro-differential equations”, Results in Applied Mathematics, 7, 1-11. google scholar
• Maturi, D. A. and Simbawa, E. A. M., 2020, The modified decomposition method for solving Volterra Fredholm integro-differential equations using maple, Int. J. GEOMATE, 18, 84-89. google scholar
• Ming, W. and Huanga, C., 2017, Collocation methods for Volterra functional integral equations with non-vanishing delays, Appl. Math Comput., 296, 198-214. google scholar
• Mishra V. N., Marasi, H. R., Shabanian H. and Sahlan M. N., 2017, Solution of Voltra Fredholm integro-differential equations using Chebyshev collocation method., Global Journal Technology and Optimization, 2, 1-4. google scholar
• Ogunniran M. O, Tiajni, N. A, Adedokun K. A. and Kareem, K. O., 2022, An accurate hybrid block technique foe second order singular problems in ordinary differential equations, African Journal of Pure and Applied Sciences 3(1): 144-154. google scholar
• Ogunrinde R. B., Obayomi, A. A. and Olayemi K. S., 2023, Numerical Solutions of third order Fredholm Integro Differential Equation VIA Linear multistep-Quadrature formulae, FUDMA, Journal of Sciences, 7(3), 33-44. google scholar
• Ogunrinde R. B., Olayemi, K. S., Isah I. O. and Salawu A. S., 2020, A numerical solver for first order initial value problems of ordinary differential equation via the combination of Chebyshev polynomial and exponential function, Journal of Physical Sciences, ISSN 2520 -084X (online), 2(1): 17-32. google scholar
• Oyedepo, T., Ishola C. Y., Ayoade A. A. and Ajileye G., 2023, Collocation computational algorithm for Volterra-Fredholm Integro-Difeerential Equations, Electronic Journal of Mathematical Analysis and Applications, 11(2), 8, 1-9. google scholar
• Oyedepo, T., Taiwo O. A., Adewale A. J., Ishaq A. A. and Ayinde A. M., 2022, Numerical Solution of System of linear fractional integro-differential equations by least squares collocation Chebyshev technique, Mathematics and Computational Sciences, 3(2): 10-21. google scholar
• Ramadan, M., Raslan, K., Hadhoud A. and Nassar M., 2016, Numerical solution of high-order linear integro-differential equations with variable coefficients using two proposed schemes for rational Chebyshev functions, New Trends in Mathematical Sciences, 4(3): 22-35. google scholar
• Sabzevari, M., 2019, A review on "Numerical Solution of nonlinear Volterra-Fredholm integral equations using hybrid of . . .” Alex Eng J., 58, 1099-1102. google scholar
• Scathar M. H. A., Rasede A. F. N., Ahmedov A. A. and Bachok, N., 2020, Numerical Solution of Nonlinear Fredholm and Volterra integrals by Newton-Kantorowich and Haar Wavelets Methods, Symmetry 12, 2032. google scholar
• Shang, X. and Han, D., 2010, Application of the variational iteration method for solving nth-order integro-differential equations, J. Comput. Appl. Math, 234, 1442-1447. google scholar
• Tunc, C. Tunc, 2021, On the stability integrability and boundedness analyses of systems of integro-differential equations with time delay retardation, Reo Real Acad. Ciene. Exactas. Fiscas Nat. Sci. A. Math., 115. google scholar
• Wazwaz A. M., 2011, Linear and Nonlinear Integral Equations Method and Applications, Higher Education Press, Beijing and Springer- Verlag Berlin Heiberg. google scholar
• Yuksel G., Gulsu and Sezer M., 2012, A Chebyshev polynomial approach for high-order linear Fredholm Volterra integro-diferential equations, Gazi University Journal of Sciences, 25(2): 393-401. google scholar