Approach For Fractional Differential Equations Research Articles

Corrigendum: New numerical approach for fractional differential equations

Corrigendum: New numerical approach for fractional differential equations

Time Dimensional Consistency Aware Analysis of Voltage Mode and Current Mode Active Fractional Circuits

In this research, the analysis of the active fractional circuits has been performed by using the fractional differential equation approach. Both voltage and current mode circuits have been taken into account. The fractional time component parameters have been included in the derivative terms within the fractional differential equations. This is because the consistency in time dimension between the fractional derivative and the conventional one which is also related to the physical measurability, is concerned. The fractional derivatives have been interpreted in Caputo sense. The resulting analytical solutions of the time dimensional consistency aware fractional differential equations have been determined. We have found that the dimensional consistency between both sides of the equations of the solutions which cannot be achieved in the previous works, can be obtained. By applying different source terms to the obtained analytical solutions, the response of both voltage and current mode circuits have been determined and the behaviours of the circuits have been analysed. The fractional time constant and pole locations in the F-plane of these circuits have been determined. Their dynamic behaviours, stabilities have been analysed. Moreover, the discussion on circuit realizations with fractional capacitor has also been made.

Two step Adams Bashforth method for time fractional Tricomi equation with non-local and non-singular Kernel

Abstract Recently, Atangana and Baleanu (AB) introduced a new fractional differentiation concept using non-local and non-singular kernel. Later on, theoretical applications, more practical applications and new numerical methods was established for solving partial differential equations in the meaning of AB derivative. In this study, the new numerical scheme was formulated by Owolabi and Atangana [A. Atangana, K.M.Owolabi, New numerical approach for fractional differential equations. Mathematical Modelling of Natural Phenomena 13.1 (2018) 1–19.] is considered for solving fractional Tricomi equation which involves the Mittag- Leffler kernel. A novel two-step Adams-Bashforth scheme is applied for the approximation of the AB fractional derivative. Stability of the numerical scheme is examined with the help of von Neumann stability analysis and induction principle. To test the applicability and suitability of the proposed method, two notable examples are considered with numerical results presented for some fractional order values.

The Time Dimensional Measurability Aware FDE Based Analysis of Active Circuit in The Fractional Domain

In this research, the analysis of the active circuit in the fractional domain has been performed by using the fractional differential equation approach where themeasurability in time dimension of the derivative term has also been concerned unlike the previous work. The OTA-C filter has been adopted as the candidate active circuit due to its compactness and renown. The derivative term of the fractional differential equation which includes the fractional time component parameters for obtaining such time dimensional measurability, has been interpreted in Caputo sense and the analytical solution of such equation has been determined with the aid of Laplace transformation. With the obtained solution, the time dimensional measurability aware fractional derivative based circuit responses to various inputs have been determined, the circuit fractional time constant and other crucial time parameters has been determined and thetemporal behaviour of the circuit has been analysed in the fractional domain. The loci of the pole on W-plane has been determined and the stability analysis has also been presented. Moreover, we also mathematically proof that the OTA-C filter in the fractional domain can be electronically realized with time dimensionalmeasurability awareness by using the state of the art fractional capacitor.

Neglecting nonlocality leads to unreliable numerical methods for fractional differential equations

In the paper titled “New numerical approach for fractional differential equations” by Atangana and Owolabi (2018) [1], it is presented a method for the numerical solution of some fractional differential equations. The numerical approximation is obtained by using just local information and the scheme does not present a memory term; moreover, it is claimed that third-order convergence is surprisingly obtained by simply using linear polynomial approximations. In this note we show that methods of this kind are not reliable and lead to completely wrong results since the nonlocal nature of fractional differential operators cannot be neglected. We illustrate the main weaknesses in the derivation and analysis of the method in order to warn other researchers and scientist to overlook this and other methods devised on similar basis and avoid their use for the numerical simulation of fractional differential equations.

New numerical approach for fractional differential equations

In the present case, we propose the correct version of the fractional Adams-Bashforth methods which take into account the nonlinearity of the kernels including the power law for the Riemann-Liouville type, the exponential decay law for the Caputo-Fabrizio case and the Mittag-Leffler law for the Atangana-Baleanu scenario.The Adams-Bashforth method for fractional differentiation suggested and are commonly use in the literature nowadays is not mathematically correct and the method was derived without taking into account the nonlinearity of the power law kernel. Unlike the proposed version found in the literature, our approximation, in all the cases, we are able to recover the standard case whenever the fractional powerα= 1. Numerical results are finally given to justify the effectiveness of the proposed schemes.

Generalized Lucas polynomial sequence approach for fractional differential equations

This article is interested in presenting and implementing two new numerical algorithms for solving multi-term fractional differential equations. The idea behind the proposed algorithms is based on establishing a novel operational matrix of fractional-order differentiation of generalized Lucas polynomials in the Caputo sense. This operational matrix serves as a powerful tool for obtaining the desired numerical solutions. The resulting solutions are spectral, and they are built on utilizing tau and collocation methods. A new treatment of convergence and error analysis of the suggested generalized Lucas expansion is presented. The presented numerical results demonstrate the efficiency, applicability and high accuracy of the proposed algorithms.

Jacobian-predictor-corrector approach for fractional differential equations

We present a novel predictor-corrector method, called Jacobian-predictor-corrector approach, for the numerical solutions of fractional ordinary differential equations, which are based on the polynomial interpolation and the Gauss-Lobatto quadrature w.r.t. the Jacobi-weight function $\omega (s)=(1-s)^{\alpha -1} (1+s)^{0}$ . This method has the computational cost O(N E ) and the convergent order N I , where N E and N I are, respectively, the total computational steps and the number of used interpolation points. The detailed error analysis is performed, and the extensive numerical experiments confirm the theoretical results and show the robustness of this method.

Short memory principle and a predictor–corrector approach for fractional differential equations

Fractional differential equations are increasingly used to model problems in acoustics and thermal systems, rheology and modelling of materials and mechanical systems, signal processing and systems identification, control and robotics, and other areas of application. This paper further analyses the underlying structure of fractional differential equations. From a new point of view, we apprehend the short memory principle of fractional calculus and farther apply a Adams-type predictor–corrector approach for the numerical solution of fractional differential equation. And the detailed error analysis is presented. Combining the short memory principle and the predictor–corrector approach, we gain a good numerical approximation of the true solution of fractional differential equation at reasonable computational cost. A numerical example is provided and compared with the exact analytical solution for illustrating the effectiveness of the short memory principle.