Enhanced numerical solutions of fractional differential equations via fibonacci neural networks Kushal Dhar Dwivedi, Anup Singh, Anirban Majumdar, J. F. Gómez-Aguilar, José R. Razo-Hernández International Journal of Modeling Simulation and Scientific Computing, 2025 The authors of this paper introduce Fibonacci Neural Networks (FNNs) as a novel approach for solving Differential Equations (DE) of any order. The FNN is structured with input, hidden, and output layers, where Fibonacci polynomials of various degrees are utilized as activation functions in the hidden layer. The trial solution of the differential equation is represented as the output of the FNN, with adjustable parameters (weights) that are refined through backpropagation during the network’s training process. To evaluate the effectiveness of this method, seven differential problems with known exact solutions are solved, allowing for a thorough assessment of accuracy. The proposed FNN approach is then compared with existing techniques, revealing higher accuracy and improved performance in solving the given problems. After verifying the accuracy of the proposed method, it was utilized to simulate the motion of an object attached to a spring within a Newtonian fluid. Furthermore, the effects of varying parameters on this motion were examined. The sensitivity of the FNN to different hyperparameters was also analyzed.
Parameter-uniform numerical method for singularly perturbed 2-D parabolic convection–diffusion problem with interior layers Anirban Majumdar, Srinivasan Natesan Mathematical Methods in the Applied Sciences, 2022 In this article, we devise a uniformly convergent numerical scheme for solving singularly perturbed two‐dimensional parabolic convection–diffusion problem with non‐smooth convection coefficients and source term. The solution of this kind of problem typically exhibits interior layers due to the discontinuity of convection coefficients and source term. To capture the interior layers, the piecewise‐uniform mesh is used in the spatial directions and the uniform mesh is considered in temporal direction. To discretize the temporal and spatial derivatives, we apply an alternating direction method and upwind method, respectively. Theoretically, we prove that the proposed method is ε‐uniformly convergent. Numerical results are presented to demonstrate the theoretical estimates.
A higher-order hybrid numerical scheme for singularly perturbed convection-diffusion problem with boundary and weak interior layers Anirban Majumdar, Srinivasan Natesan International Journal of Mathematical Modelling and Numerical Optimisation, 2020 In this paper, we study the numerical solutions of singularly perturbed convection-diffusion two-point BVP as well as one-dimensional parabolic convection-diffusion IBVP with discontinuous convection coefficient (positive throughout the domain) and source term. The analytical solutions of these kind of problems exhibit a boundary layer near x = 0 and a weak interior layer near x = ξ. We discretise the spatial domain by the piecewise-uniform Shishkin mesh and the temporal domain by a uniform mesh. To approximate the spatial derivatives, we apply the hybrid finite difference scheme. The implicit-Euler scheme is used for discretising the temporal derivative. For the time independent problem, we derive that the proposed hybrid scheme is ε-uniformly convergent of almost second-order and for the time dependent problem, we also prove that the proposed scheme is ε-uniformly convergent of almost second-order in space and first-order in time. To validate the theoretical estimates, some numerical results are presented.
An ϵ-uniform hybrid numerical scheme for a singularly perturbed degenerate parabolic convection–diffusion problem Anirban Majumdar, Srinivasan Natesan International Journal of Computer Mathematics, 2019 In this paper, we study the numerical solution of singularly perturbed degenerate parabolic convection–diffusion problem on a rectangular domain. The solution of the problem exhibits a parabolic boundary layer in the neighbourhood of x=0. First, we use the backward-Euler finite difference scheme to discretize the time derivative of the continuous problem on uniform mesh in the temporal direction. Then, to discretize the spatial derivatives of the resulting time semidiscrete problem, we apply the hybrid finite difference scheme, which is a combination of central difference scheme and midpoint upwind scheme on piecewise uniform Shishkin mesh. We derive the error estimates, which show that the proposed hybrid scheme is ϵ-uniform convergent of almost second-order (up to a logarithmic factor) in space and first-order in time. Some numerical results have been carried out to validate the theoretical results.
Enhanced numerical solutions of fractional differential equations via fibonacci neural networks KD Dwivedi, A Singh, A Majumdar, JF Gómez-Aguilar, ... International Journal of Modeling, Simulation, and Scientific Computing 16 … , 2025 2025 Citations: 1
Numerical scheme for partial differential equations involving small diffusion term with non-local boundary conditions S Bala, L Govindarao, A Das, A Majumdar Journal of Applied Mathematics and Computing 69, 4307–4331 , 2023 2023 Citations: 4
Parameter‐uniform numerical method for singularly perturbed 2‐D parabolic convection–diffusion problem with interior layers A Majumdar, S Natesan Mathematical Methods in the Applied Sciences 45 (5), 3039-3057 , 2022 2022 Citations: 3
A higher-order hybrid numerical scheme for singularly perturbed convection-diffusion problem with boundary and weak interior layers A Majumdar, S Natesan International Journal of Mathematical Modelling and Numerical Optimisation … , 2020 2020 Citations: 10
An ϵ -uniform hybrid numerical scheme for a singularly perturbed degenerate parabolic convection–diffusion problem A Majumdar, S Natesan International Journal of Computer Mathematics 96 (7), 1313-1334 , 2019 2019 Citations: 36
Second-order uniformly convergent Richardson extrapolation method for singularly perturbed degenerate parabolic PDEs A Majumdar, S Natesan International Journal of Applied and Computational Mathematics 3 (Suppl 1 … , 2017 2017 Citations: 29
Alternating direction numerical scheme for singularly perturbed 2D degenerate parabolic convection-diffusion problems A Majumdar, S Natesan Applied Mathematics and Computation 313, 453-473 , 2017 2017 Citations: 24
Robust numerical methods for singularly perturbed parabolic PDEs with interior and boundary layers A Majumdar 2017
MOST CITED SCHOLAR PUBLICATIONS
An ϵ -uniform hybrid numerical scheme for a singularly perturbed degenerate parabolic convection–diffusion problem A Majumdar, S Natesan International Journal of Computer Mathematics 96 (7), 1313-1334 , 2019 2019 Citations: 36
Second-order uniformly convergent Richardson extrapolation method for singularly perturbed degenerate parabolic PDEs A Majumdar, S Natesan International Journal of Applied and Computational Mathematics 3 (Suppl 1 … , 2017 2017 Citations: 29
Alternating direction numerical scheme for singularly perturbed 2D degenerate parabolic convection-diffusion problems A Majumdar, S Natesan Applied Mathematics and Computation 313, 453-473 , 2017 2017 Citations: 24
A higher-order hybrid numerical scheme for singularly perturbed convection-diffusion problem with boundary and weak interior layers A Majumdar, S Natesan International Journal of Mathematical Modelling and Numerical Optimisation … , 2020 2020 Citations: 10
Numerical scheme for partial differential equations involving small diffusion term with non-local boundary conditions S Bala, L Govindarao, A Das, A Majumdar Journal of Applied Mathematics and Computing 69, 4307–4331 , 2023 2023 Citations: 4
Parameter‐uniform numerical method for singularly perturbed 2‐D parabolic convection–diffusion problem with interior layers A Majumdar, S Natesan Mathematical Methods in the Applied Sciences 45 (5), 3039-3057 , 2022 2022 Citations: 3
Enhanced numerical solutions of fractional differential equations via fibonacci neural networks KD Dwivedi, A Singh, A Majumdar, JF Gómez-Aguilar, ... International Journal of Modeling, Simulation, and Scientific Computing 16 … , 2025 2025 Citations: 1
Robust numerical methods for singularly perturbed parabolic PDEs with interior and boundary layers A Majumdar 2017
