Showing papers by "Pushpendra Kumar published in 2023"
TL;DR: In this article , a new numerical method to solve fractional differential equations containing Caputo-Fabrizio derivatives is presented. But the method is not suitable for solving fixed point problems, and error and stability analyses are performed briefly.
Abstract: In this article, we derive a new numerical method to solve fractional differential equations containing Caputo-Fabrizio derivatives. The fundamental concepts of fractional calculus, numerical analysis, and fixed point theory form the basis of this study. Along with the derivation of the algorithm of the proposed method, error and stability analyses are performed briefly. To explore the validity and effectiveness of the proposed method, several examples are simulated, and the new solutions are compared with the outputs of the previously published two-step Adams-Bashforth method.
6 citations
TL;DR: In this paper , a new version of L1-Predictor-Corrector (L1-PC) method was derived by using some previously given methods (L 1-PC for single delay, PC for non-delay, and decomposition algorithm) to solve multiple delay-type fractional differential equations.
Abstract: In this paper, we derive a new version of L1-Predictor–Corrector (L1-PC) method by using some previously given methods (L1-PC for single delay, PC for non-delay, and decomposition algorithm) to solve multiple delay-type fractional differential equations. The Caputo fractional derivative with singular type kernel is used to establish the results. Some important remarks related to the delay term estimation and error analysis are mentioned. In order to check the accuracy and correctness of our method, we solve a neural network system with two delay parameters. A number of graphs are given to justify the role of delays as well as the accuracy of the algorithm. The given method is fully novel and reliable to solve multiple delay type fractional order systems in Caputo sense.
3 citations
2 citations
TL;DR: In this paper , the coefficients of fractional Adams-Bashforth and Adams-Moulton methods were used to construct higher-order fractional linear multi-step methods in general form, with extended stability regions.
Abstract: In this paper, we propose two arrays, containing the coefficients of fractional Adams-Bashforth and Adams-Moulton methods, and also recursive relations to produce the elements of these arrays. Then, we illustrate the application of these arrays in a suitable way to construct higher-order fractional linear multi-step methods in general form, with extended stability regions. The effectiveness of the new method is shown in comparison with some available previous results in an illustrative test problem.
1 citations
TL;DR: In this article , a neural network-based approach with an Extreme Learning Machine (ELM) for solving fractional differential equations has been proposed and the convergence and stability of the proposed method is provided.
1 citations
01 Jan 2023
TL;DR: In this paper , a nonlinear Caputo-type snakebite envenoming model with memory was introduced, and the numerical solution of the model was derived from a novel implementation of a finite-difference predictor-corrector (L1-PC) scheme with error estimation and stability analysis.
Abstract: In this article, we introduce a nonlinear Caputo-type snakebite envenoming model with memory. The well-known Caputo fractional derivative is used to generalize the previously presented integer-order model into a fractional-order sense. The numerical solution of the model is derived from a novel implementation of a finite-difference predictor-corrector (L1-PC) scheme with error estimation and stability analysis. The proof of the existence and positivity of the solution is given by using the fixed point theory. From the necessary simulations, we justify that the first-time implementation of the proposed method on an epidemic model shows that the scheme is fully suitable and time-efficient for solving epidemic models. This work aims to show the novel application of the given scheme as well as to check how the proposed snakebite envenoming model behaves in the presence of the Caputo fractional derivative, including memory effects.
TL;DR: In this paper , a hybrid prediction method known as the Grey-Fourier Markov model which includes Grey models (GM), Fourier series, and Markov state transition was proposed to increase the forecasting accuracy.
Abstract: In the history of the gold market, contemporary gold prices are higher than the previous values, and the current gold market is highly non-linear and unpredictable. Gold is the most popular precious metal for investment out of all the precious metals. The gold market, like other markets, is vulnerable to speculation and volatility. Gold has served as a secure base in several countries when compared to other precious metals used for investment. In this study, we suggest a time series model for predicting daily variations in the amount of gold per gram in Indian rupees. To increase the forecasting accuracy, we use a hybrid prediction method known as the Grey-Fourier Markov model which includes Grey models (GM), Fourier series, and Markov state transition. Here, we divide the forecasting process into three steps. The first step is to simulate the data of daily volatile price of gold using GM (1, 1), GM (2, 1), and Grey Verhulst models and also to calculate corresponding residual errors. In the second step, we utilize the residual error produced by the above grey models to predict the trend of the gold price with the help of the Fourier series and Markov Model. In the third step, we use hybrid grey models to improve the precision. Finally, we conclude that the proposed methodology outperforms the aforementioned strategies in terms of results.
TL;DR: In this article , a fractional order nonlinear model for Omicron, known as B.1.529 SARS-Cov-2 variant, is proposed, and the COVID-19 vaccine and quarantine are inserted to ensure the safety of host population in the model.
Abstract: In this paper, a fractional order nonlinear model for Omicron, known as B.1.1.529 SARS-Cov-2 variant, is proposed. The COVID-19 vaccine and quarantine are inserted to ensure the safety of host population in the model. The fundamentals of positivity and boundedness of the model solution are simulated. The reproduction number is estimated to determine whether or not the epidemic will spread further in Tamilnadu, India. Real Omicron variant pandemic data from Tamilnadu, India, are validated. The fractional-order generalization of the proposed model, along with real data-based numerical simulations, is the novelty of this study.
TL;DR: In this article , the existence, uniqueness, and stability of generalized Caputo-type fractional boundary value problems (FBVPs) were established using the Banach contraction principle along with necessary features of fixed point theory.
Abstract: In this article, we derive some novel results of the existence, uniqueness, and stability of the solution of generalized Caputo-type fractional boundary value problems (FBVPs). The Banach contraction principle, along with necessary features of fixed point theory, is used to establish our results. An example is illustrated to justify the validity of the theoretical observations.
TL;DR: In this article , an effective numerical method using two-dimensional shifted fractional-order Gegenbauer multi-wavelets to find the approximate solutions of the time-fractional distributed order nonlinear partial differential equations was proposed.
Abstract: In this paper, we propose an effective numerical method using two-dimensional Shifted fractional-order Gegenbauer Multi-wavelets to find the approximate solutions of the time-fractional distributed order non-linear partial differential equations. The method is applied to numerically solve the fractional distributed order non-linear Klein–Gordon equation. We derive an exact formula for the Riemann-Liouville fractional integral operator for the Shifted fractional Gegenbauer Multi-wavelets. Applying function approximations obtained by this method turns the considered equation into a system of algebraic equations. Error estimation and convergence analysis of the method are also studied. Some numerical examples are included to show and check the effectiveness of the proposed method.
TL;DR: In this paper , a neural network-based approach was proposed for solving generalized Caputo-type fractional differential equations with and without delay by coupling the theory of functional connections and a new loss function.
TL;DR: In this article , the authors solved a model of a well-known infectious disease called dengue fever via fractional natural decomposition and modified Predictor-Corrector (PC) methods.
Abstract: In this paper, we solved a model of a well-known infectious disease called dengue fever via fractional natural decomposition and modified Predictor–Corrector (PC) methods. A study of the dengue epidemic in the Cape Verde Islands off the coast of West Africa in 2009 has been resumed here for a better understanding of the results. The results are obtained using Liouville–Caputo and new generalized Caputo-type fractional derivatives. The numerical simulations are presented for various orders of given derivatives. Existence and uniqueness analysis of the given problem are also performed in the new generalized Caputo sense. The explored results are verified using figures. The main target of this paper is to explore the different dynamics of the given dengue fever model via two types of fractional numerical algorithms.
TL;DR: In this paper , the authors considered a fractional-order neuron model under an electromagnetic field in terms of generalized Caputo fractional derivatives and numerically solved the problem by using a generalized version of the Euler method with stability and error analysis.
Abstract: Abstract This article considers a fractional-order neuron model under an electromagnetic field in terms of generalized Caputo fractional derivatives. The motivation for incorporating fractional derivatives in the previously proposed integer-order neuron model is that the fractional-order model impresses with efficient effects of the memory, and parameters with fractional orders can increase the model performance by amplifying a degree of freedom. The results on the uniqueness of the solution for the proposed neuron model are established using well-known theorems. The given model is numerically solved by using a generalized version of the Euler method with stability and error analysis. Several graphical simulations are performed to capture the variations in the membrane potential considering no electromagnetic field effects, various frequency brands of external forcing current, and the amplitude and frequency of the external magnetic radiation. The impacts of fractional-order cases are clearly justified.