Fibonacci Collocation Method for Solving a Class of Nonlinear Differential Equations

Year 2023, , 42 - 53, 30.06.2023

Musa Çakmak , Sertan Alkan

https://doi.org/10.47000/tjmcs.960168

Abstract

In this study, a collocation method based on Fibonacci polynomials is used for approximately solving a class of nonlinear differential equations with initial conditions. The problem is firstly reduced into a nonlinear algebraic system via collocation points, later the unknown coefficients of the approximate solution function are calculated. Also, some problems are presented to test the performance of the proposed method by using error functions. Additionally, the obtained numerical results are compared with exact solutions of the test problems and approximate ones obtained with other methods in literature.

Keywords

Fibonacci collocation method, nonlinear differential equations, initial value problems

References

• Akgul, A., Hashemi, M.S., Inc, M., Raheem, S.A., Constructing two powerful methods to solve the Thomas-Fermi equation, Nonlinear Dynamics, 87(2)(2016), 1435–1444.
• Akgul, A., Inc, M., Hashemi, M.S., Group preserving scheme and reproducing kernel method for the Poisson-Boltzmann equation for semiconductor devices, Nonlinear Dynamics, 88(4)(2017), 2817–2829.
• Chaharborj, S.S., See, P.P., Application of Chebyshev neural network to solve Van der Pol equations, International Journal of Basic and Applied Sciences, 10(1)(2021), 7–19.
• Falcon, S., Plaza, A., The k-Fibonacci sequence and the Pascal 2-triangle, Chaos, Solitons & Fractals, 33(1)(2007), 38–49.
• Falcon, S., Plaza, A., On k-Fibonacci sequences and polynomials and their derivatives, Chaos, Solitons & Fractals, 39(3)(2009), 1005–1019.
• Gospodarczyk, P., Wozny, P., An iterative approximate method of solving boundary value problems using dual Bernstein polynomials, Tech. Rep. 2018-03-01, University of Wroclaw Institute of Computer Science, arXiv:1709.02162 (2018).
• Guler, C., A new numerical algorithm for the Abel equation of the second kind, International Journal of Computer Mathematics, 84(1)(2007), 109–119.
• Gulsu, M., Ozturk, Y., Sezer, M., On the solution of the Abel equation of the second kind by the shifted Chebyshev polynomials, Applied Mathematics and Computation, 217(9)(2011), 4827–4833.
• Hashemi, M.S., Constructing a new geometric numerical integration method to the nonlinear heat transfer equations, Communications in Nonlinear Science and Numerical Simulation, 22(1-3)(2015), 990–1001.
• Hashemi, M.S., Akgul, A., On the MHD boundary layer flow with diffusion and chemical reaction over a porous flat plate with suction/blowing: two reliable methods, Engineering with Computers, 37(2)(2019), 1147–1158.
• Hashemi, M.S., Baleanu, D., Numerical approximation of higher-order time-fractional telegraph equation by using a combination of a geometric approach and method of line, Journal of Computational Physics, 316(2016), 10–20.
• Hashemi, M.S., Inc, M., Hajikhah, S., Generalized squared remainder minimization method for solving multi-term fractional differential equations, Nonlinear Analysis: Modelling and Control, 26(1)(2021), 57–71.
• He, J.H., Latifizadeh, H., A general numerical algorithm for nonlinear dierential equations by the variational iteration method, International Journal of Numerical Methods for Heat & Fluid Flow, 30(11)(2020), 4797–4810.
• Kurt, A., Yalcinbas, S., Sezer, M., Fibonacci collocation method for solving linear differential-difference equations, Mathematical and Computational Applications, 18(3)(2013), 448–458.
• Kurt, A., Yalcinbas, S., Sezer, M., Fibonacci collocation method for solving high-order linear Fredholm integro-differential-difference equations, International Journal of Mathematics and Mathematical Sciences, 2013(2013), 1–9.
• Mirzaee, F., Hoseini, S.F., Solving systems of linear Fredholm integro-differential equations with Fibonacci polynomials, Ain Shams Engineering Journal, 5(1)(2014), 271–283.
• Mirzaee, F., Hoseini, S.F., Solving singularly perturbed differential-difference equations arising in science and engineering with Fibonacci polynomials, Results in Physics, 3(2013), 134–141.
• Mirzaee, F., Hoseini, S.F., A new collocation approach for solving systems of high-order linear Volterra integro-differential equations with variable coeffcients, Applied Mathematics and Computation, 311(2017), 272–282.
• Odibat, Z., An optimized decomposition method for nonlinear ordinary and partial dierential equations, Physica A: Statistical Mechanics and its Applications, 541(2020), 123323.
• Odibat, Z., An improved optimal homotopy analysis algorithm for nonlinear dierential equations, Journal of Mathematical Analysis and Applications, 488(2)(2020), 124089.
• Rani, D., Mishra, V., Numerical inverse Laplace transform based on Bernoulli polynomials operational matrix for solving nonlinear differential equations, Results in Physics, 16(2020), 102836.
• Simao, P.D., Post-buckling analysis of nonlinear shear-deformable prismatic columns using a GBT consistent energy formulation, Computers and Structures, 190(2017), 186–204.