Nagumo equation is a nonlinear diffusion reaction equation usually used to model the transmission of nerve impulse, logistic population growth, branching Brownian motion process etc. This work explores the utility âŚ
Nagumo equation is a nonlinear diffusion reaction equation usually used to model the transmission of nerve impulse, logistic population growth, branching Brownian motion process etc. This work explores the utility of a sixth order compact finite difference scheme in space and low-storage third order total variation diminishing Runge-Kutta (TVD-RK3) scheme in time combined to find the numerical solution of Nagumo equation. The comparison of the computed numerical results with the exact solution revealed that the proposed method to solve Nagumo equation is capable of achieving higher accuracy than some existing compact finite difference methods.
Nagumo equation is a nonlinear diffusion reaction equation usually used to model the transmission of nerve impulse, logistic population growth, branching Brownian motion process etc. This work explores the utility âŚ
Nagumo equation is a nonlinear diffusion reaction equation usually used to model the transmission of nerve impulse, logistic population growth, branching Brownian motion process etc. This work explores the utility of a sixth order compact finite difference scheme in space and low-storage third order total variation diminishing Runge-Kutta (TVD-RK3) scheme in time combined to find the numerical solution of Nagumo equation. The comparison of the computed numerical results with the exact solution revealed that the proposed method to solve Nagumo equation is capable of achieving higher accuracy than some existing compact finite difference methods.
The FitzHugh-Nagumo equation is an important nonlinear reaction-diffusion equation used in physics and chemicals. To obtain the numerical solution of partial differential equations, the compact finite difference method is widely âŚ
The FitzHugh-Nagumo equation is an important nonlinear reaction-diffusion equation used in physics and chemicals. To obtain the numerical solution of partial differential equations, the compact finite difference method is widely applied. In this paper, I propose a new numerical solution to FitzHugh-Nagumo equation by using a fourth-order compact finite difference scheme in space, and a semi-implicit Crank-Nicholson method in time. I further calculate the results in terms of accuracy by leveraging the proposed method and exact solution. In particular, I compare the new method whose convergence order is close to four with the second order central difference method. The simulated results show the new solution is more accurate and effective. The proposed method is expected to be a good solution to some problems in the real world.
In this work, the homotopy perturbation method (HPM), the variational iteration method (VIM) and the Adomian decomposition method (ADM) are applied to solve the FitzhughâNagumo equation. Numerical solutions obtained by âŚ
In this work, the homotopy perturbation method (HPM), the variational iteration method (VIM) and the Adomian decomposition method (ADM) are applied to solve the FitzhughâNagumo equation. Numerical solutions obtained by these methods when compared with the exact solutions reveal that the obtained solutions produce high accurate results. The results show that the HPM, the VIM and the ADM are of high accuracy and are efficient for solving the FitzhughâNagumo equation. Also the results demonstrate that the introduced methods are powerful tools for solving the nonlinear partial differential equations. Copyright Š 2010 John Wiley & Sons, Ltd.
In this work, we describe the radial basis functions for solving the time fractional partial differential equations defined by Caputo sense. These problems can be discretized in the time direction âŚ
In this work, we describe the radial basis functions for solving the time fractional partial differential equations defined by Caputo sense. These problems can be discretized in the time direction based on finite difference scheme and is continuously approximated by using the radial basis functions in the space direction which achieves the semi-discrete solution. Numerical results accuracy the efficiency of the presented method.
In this paper, we approximate the solution of the initial and boundary value problems of anomalous second- and fourth-order sub-diffusion equations of fractional order. The fractional derivative is used in âŚ
In this paper, we approximate the solution of the initial and boundary value problems of anomalous second- and fourth-order sub-diffusion equations of fractional order. The fractional derivative is used in the Caputo sense. To solve these equations, we will use a numerical method based on B-spline basis functions and the collocation method. It will be shown that the proposed scheme is unconditionally stable and convergent. Three numerical examples are adopted to demonstrate the performance of the proposed scheme.
In this paper, a reliable implicit di erence scheme is proposed to analyze the fractional fourth-order subdi usion equation on a bounded domain. The time-fractional derivative operator is characterized in âŚ
In this paper, a reliable implicit di erence scheme is proposed to analyze the fractional fourth-order subdi usion equation on a bounded domain. The time-fractional derivative operator is characterized in the Ji Huan He's sense, and the space derivative is approximated by the ve-point centered formula. The numerical parameters, i.e. consistency, stability, and convergence analyses of the considered scheme, are proven.
One of the most widely studied biological systems with excitable behavior is neural communication by nerve cells via electrical signaling. The FitzhughâNagumo equation is a simplification of the HodginâHuxley model âŚ
One of the most widely studied biological systems with excitable behavior is neural communication by nerve cells via electrical signaling. The FitzhughâNagumo equation is a simplification of the HodginâHuxley model (Hodgin and Huxley, 1952) [24] for the membrane potential of a nerve axon. In this paper we developed a three time-level implicit method by using tension spline function. The resulting equations are solved by a tri-diagonal solver. We described the mathematical formulation procedure in detail. The stability of the presented method is investigated. Results of numerical experiments verify the theoretical behavior of the orders of convergence.
Abstract The key objective of this paper is to study the fractional model of Fitzhugh-Nagumo equation (FNE) with a reliable computationally effective numerical scheme, which is compilation of homotopy perturbation âŚ
Abstract The key objective of this paper is to study the fractional model of Fitzhugh-Nagumo equation (FNE) with a reliable computationally effective numerical scheme, which is compilation of homotopy perturbation method with Laplace transform approach. Homotopy polynomials are employed to simplify the nonlinear terms. The convergence and error analysis of the proposed technique are presented. Numerical outcomes are shown graphically to prove the efficiency of proposed scheme.
In this manuscript we proposed a new fractional derivative with non-local and no-singular kernel. We presented some useful properties of the new derivative and applied it to solve the fractional âŚ
In this manuscript we proposed a new fractional derivative with non-local and no-singular kernel. We presented some useful properties of the new derivative and applied it to solve the fractional heat transfer model.
In this paper, we developed the compact finite differences method to find approximate solutions for the FitzHugh-Nagumo (F-N) equations. To the best of our knowledge, until now there is no âŚ
In this paper, we developed the compact finite differences method to find approximate solutions for the FitzHugh-Nagumo (F-N) equations. To the best of our knowledge, until now there is no compact finite difference solutions have been reported for the FitzHugh-Nagumo equation arising in gene propagation and model. We have given numerical example to demonstrate the validity and applicability.
Starting with an introduction to fractional derivatives and numerical approximations, this book presents finite difference methods for fractional differential equations, including time-fractional sub-diffusion equations , time-fractional wave equations, and space-fractional âŚ
Starting with an introduction to fractional derivatives and numerical approximations, this book presents finite difference methods for fractional differential equations, including time-fractional sub-diffusion equations , time-fractional wave equations, and space-fractional differential equations, among others. Approximation methods for fractional derivatives are developed and approximate accuracies are analyzed in detail.
Abstract This study presents a robust modification of Chebyshev Ď âweighted CrankâNicolson method for analyzing the subâdiffusion equations in the Caputo fractional sense. In order to solve the problem, by âŚ
Abstract This study presents a robust modification of Chebyshev Ď âweighted CrankâNicolson method for analyzing the subâdiffusion equations in the Caputo fractional sense. In order to solve the problem, by discretization of the subâfractional diffusion equations using Taylor's expansion a linear system of algebraic equations that can be analyzed by numerical methods is presented. Furthermore, consistency, convergence, and stability analysis of the suggested method are discussed. In this framework, compact structures of subâdiffusion equations are considered as prototype examples. The main advantage of the proposed method is that, it is more efficient in terms of CPU time, computational cost and accuracy in comparing with the existing ones in open literature.
Nagumo equation is a nonlinear diffusion reaction equation usually used to model the transmission of nerve impulse, logistic population growth, branching Brownian motion process etc. This work explores the utility âŚ
Nagumo equation is a nonlinear diffusion reaction equation usually used to model the transmission of nerve impulse, logistic population growth, branching Brownian motion process etc. This work explores the utility of a sixth order compact finite difference scheme in space and low-storage third order total variation diminishing Runge-Kutta (TVD-RK3) scheme in time combined to find the numerical solution of Nagumo equation. The comparison of the computed numerical results with the exact solution revealed that the proposed method to solve Nagumo equation is capable of achieving higher accuracy than some existing compact finite difference methods.
The aim of this paper is to put on display the numerical solutions and dynamics of time fractional Fitzhugh-Nagumo model, which is an important nonlinear reaction-diffusion equation. For this purpose, âŚ
The aim of this paper is to put on display the numerical solutions and dynamics of time fractional Fitzhugh-Nagumo model, which is an important nonlinear reaction-diffusion equation. For this purpose, finite element method based on trigonometric cubic B-splines are used to obtain numerical solutions of the model. In this model, the derivative which is fractional order is taken in terms of Caputo. Thus, time dicretization is made using L1L1 algorithm for Caputo derivative and space discretization is made using trigonometric cubic B- spline basis. Also, the non-linear term in the model is linearized by the Rubin Graves type linearization. The error norms are calculated for measuring the accuracy of the finite element method. The comparison of numerical and exact solutions are exhibited via tables and graphics.
This study mainly investigates new techniques for obtaining numerical solutions of time-fractional diffusion equations. The fractional derivative term is represented in the Lagrange operational sense. First, we describe the temporal âŚ
This study mainly investigates new techniques for obtaining numerical solutions of time-fractional diffusion equations. The fractional derivative term is represented in the Lagrange operational sense. First, we describe the temporal direction of the considered model using the Legendre orthogonal polynomials. Moreover, to archive a full discretization approach a type of nonlocal method has been applied that is known as the nonlocal peridynamic differential operator (PDDO). The PDDO is based on the concept of peridynamic (PD) interactions by proposing the PD functions orthogonal to each term in the Taylor Series Expansion (TSE) of a field variable. The PDDO for numerical integration uses the vicinity of each point (referred to as the horizon, which does not need background mesh). The PDDO is exclusively described in terms of integration (summation) throughout the interaction domain. As a result, it is unsusceptible to singularities caused by discontinuities. It does, however, need the creation of PD functions at each node. We numerically investigate the stability, the convergence of the scheme, which verifies the validity of the proposed method. Numerical results show the simplicity and accuracy of the presented method.
This paper proposes a novel nonlinear fractional-order pandemic model with Caputo derivative for corona virus disease. A nonstandard finite difference (NSFD) approach is presented to solve this model numerically. This âŚ
This paper proposes a novel nonlinear fractional-order pandemic model with Caputo derivative for corona virus disease. A nonstandard finite difference (NSFD) approach is presented to solve this model numerically. This strategy preserves some of the most significant physical properties of the solution such as non-negativity, boundedness and stability or convergence to a stable steady state. The equilibrium points of the model are analyzed and it is determined that the proposed fractional model is locally asymptotically stable at these points. Non-negativity and boundedness of the solution are proved for the considered model. Fixed point theory is employed for the existence and uniqueness of the solution. The basic reproduction number is computed to investigate the dynamics of corona virus disease. It is worth mentioning that the non-integer derivative gives significantly more insight into the dynamic complexity of the corona model. The suggested technique produces dynamically consistent outcomes and excellently matches the analytical works. To illustrate our results, we conduct a comprehensive quantitative study of the proposed model at various quarantine levels. Numerical simulations show that can eradicate a pandemic quickly if a human population implements obligatory quarantine measures at varying coverage levels while maintaining sufficient knowledge.
In this paper, we study time-fractional diffusion equations such as the time-fractional Kolmogorov equations (TFâKEs) and the time-fractional advectionâdiffusion equations (TFâADEs) in the Caputo sense. Here, we have developed the âŚ
In this paper, we study time-fractional diffusion equations such as the time-fractional Kolmogorov equations (TFâKEs) and the time-fractional advectionâdiffusion equations (TFâADEs) in the Caputo sense. Here, we have developed the operational matrices (OMs) using the Hosoya polynomial (HP) as basis function for OMs to obtain solution of the TFâKEs and the TFâADEs. The great benefit of this technique is converting the TFâKEs and the TFâADEs to algebraic equations, which can be simply solved the problem under study. We provide error bound for the approximation of a bivariate function using the HP. Furthermore, comparison of the numerical results obtained using the proposed technique with the exact solution is done. The results prove that the proposed numerical method is most relevant for solving the TFâKEs and the TFâADEs and accurate.
In the fuzzy calculus, the study of fuzzy differential equations (FDEs) created a proper setting to model real problems which contain vagueness or uncertainties factors. In this paper, we consider âŚ
In the fuzzy calculus, the study of fuzzy differential equations (FDEs) created a proper setting to model real problems which contain vagueness or uncertainties factors. In this paper, we consider a class fuzzy differential equations (FFDEs) with non-integer or variable order (VO). The variable order derivative is defined in the AtanganaâBaleanuâCaputo sense on fuzzy set-valued functions. The main problem under the fuzzy initial condition is converted to a new problem by the [Formula: see text]-cut representation of fuzzy-valued function. For solving the new problem, we use the operational matrices (OMs) based on the shifted Legendre polynomials (SLPs). By approximating the unknown function and its derivative in terms of the SLPs and substituting these approximations into the equation, the main problem is converted to a system of nonlinear algebraic equations. An error estimate of the numerical solution is proved. Finally, an example is considered to confirm the accuracy of the proposed technique.
An introduction to fractional calculus, P.L. Butzer & U. Westphal fractional time evolution, R. Hilfer fractional powers of infinitesimal generators of semigroups, U. Westphal fractional differences, derivatives and fractal time âŚ
An introduction to fractional calculus, P.L. Butzer & U. Westphal fractional time evolution, R. Hilfer fractional powers of infinitesimal generators of semigroups, U. Westphal fractional differences, derivatives and fractal time series, B.J. West and P. Grigolini fractional kinetics of Hamiltonian chaotic systems, G.M. Zaslavsky polymer science applications of path integration, integral equations, and fractional calculus, J.F. Douglas applications to problems in polymer physics and rheology, H. Schiessel et al applications of fractional calculus and regular variation in thermodynamics, R. Hilfer.
This work is focused on the derivation and analysis of a novel numerical technique for solving time fractional mobile-immobile advection-dispersion equation which models many complex systems in engineering and science. âŚ
This work is focused on the derivation and analysis of a novel numerical technique for solving time fractional mobile-immobile advection-dispersion equation which models many complex systems in engineering and science. The scheme is derived using the effective combination of Euler and Caputo numerical techniques for approximating the integer and fractional time derivatives respectively, and a fourth order exponential compact scheme for spatial derivatives. The Fourier analysis technique is used to prove that the proposed numerical scheme is unconditionally stable and perform convergence analysis. To assess the viability and accuracy of the proposed scheme, some numerical examples are demonstrated with constant as well as variable order time fractional derivatives for this model.
Recently, modeling problems in various field of sciences and engineering with the help of fractional calculus has been welcomed by researchers. One of these interesting models is a brain tumor âŚ
Recently, modeling problems in various field of sciences and engineering with the help of fractional calculus has been welcomed by researchers. One of these interesting models is a brain tumor model. In this framework, a two dimensional expansion of the diffusion equation and glioma growth is considered. The analytical solution of this model is not an easy task, so in this study, a numerical approach based on the operational matrix of conventional orthonormal Bernoulli polynomials (OBPs) has been used to estimate the solution of this model. As an important advantage of the proposed method is to obtain the fractional derivative in matrix form, which makes calculations easier. Also, by using this technique, the problem under the study is converted into a system of nonlinear algebraic equations. This system is solved via Newton's method and the error analysis is presented. At the end to show the accuracy of the work, we have examined two examples and compared the numerical results with other works.
This textbook introduces several major numerical methods for solving various partial differential equations (PDEs) in science and engineering, including elliptic, parabolic, and hyperbolic equations. It covers traditional techniques that include âŚ
This textbook introduces several major numerical methods for solving various partial differential equations (PDEs) in science and engineering, including elliptic, parabolic, and hyperbolic equations. It covers traditional techniques that include the classic finite difference method and the finite element method as well as state-of-the-art numerical