An unstructured mesh control volume method for two-dimensional space fractional diffusion equations with variable coefficients on convex domains

In this paper, we propose a novel unstructured mesh control volume method to deal with the space fractional derivative on arbitrarily shaped convex domains, which to the best of our knowledge is a new contribution to the literature. Firstly, we present the finite volume scheme for the two-dimensional space fractional diffusion equation with variable coefficients and provide the full implementation details for the case where the background interpolation mesh is based on triangular elements. Secondly, we explore the property of the stiffness matrix generated by the integral of space fractional derivative. We find that the stiffness matrix is sparse and not regular. Therefore, we choose a suitable sparse storage format for the stiffness matrix and develop a fast iterative method to solve the linear system, which is more efficient than using the Gaussian elimination method. Finally, we present several examples to verify our method, in which we make a comparison of our method with the finite element method for solving a Riesz space fractional diffusion equation on a circular domain. The numerical results demonstrate that our method can reduce CPU time significantly while retaining the same accuracy and approximation property as the finite element method. The numerical results also illustrate that our method is effective and reliable and can be applied to problems on arbitrarily shaped convex domains.Comment: 18 pages, 5 figures, 9 table

Stability and convergence based on the finite difference method for the nonlinear fractional cable equation on non-uniform staggered grids

Abstract(#br)In this article, a block-centered finite difference method for the nonlinear fractional cable equation is introduced and analyzed. The unconditional stability and the global convergence of the scheme are proved rigorously. Some a priori estimates of discrete norms with optimal order of convergence O ( Δ t α + h 2 + k 2 ) both for pressure and velocity are established on non-uniform rectangular grids, where α = min ⁡ { 1 + γ 1 , 1 + γ 2 } , Δ t , h and k are the step sizes in time, space in x - and y -direction. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis

Active vibration control of flexible beam incorporating recursive least square and neural network algorithms

In recent years, active vibration control (AVC) has emerged as an important area of scient ific study especially for vibrat ion suppression of flexible structures. Flexible structures offer great advantages in contrast to the conventional structures, but necessary action must be taken for cancelling the unwanted vibration. In this research, a simulation algorithm represent ing flexible beam with specific condit ions was derived from Euler Bernoulli beam theory. The proposed finite difference (FD) algorithm was developed in such way that it allows the disturbance excitat ion at various points. The predicted resonance frequencies were recorded and validated with theoretical and experimental values. Subsequent ly, flexible beam test rig was developed for collecting data to be used in system ident ificat ion (SI) and controller development. The experimental rig was also utilised for implementation and validat ion of controllers. In this research, parametric and nonparametric SI approaches were used for characterising the dynamic behaviour of a lightweight flexible beam using input - output data collected experimentally. Tradit ional recursive least square (RLS) method and several artificial neural network (ANN) architectures were utilised in emulat ing this highly nonlinear dynamic system here. Once the model of the system was obtained, it was validated through a number of validation tests and compared in terms of their performance in represent ing a real beam. Next, the development of several convent ional and intelligent control schemes with collocated and non-collocated actuator sensor configurat ion for flexible beam vibrat ion attenuation was carried out. The invest igat ion involves design of convent ional proportional-integral-derivat ive (PID) based, Inverse recursive least square active vibrat ion control (RLS-AVC), Inverse neuro active vibration control (Neuro-AVC), Inverse RLS-AVC with gain and Inverse Neuro-AVC with gain controllers. All the developed controllers were tested, verified and validated experimentally. A comprehensive comparat ive performance to highlight the advantages and drawbacks of each technique was invest igated analyt ically and experimentally. Experimental results obtained revealed the superiorit y of Inverse RLS-AVC with gain controller over convent ional method in reducing the crucial modes of vibration of flexible beam structure. Vibration attenuation achieved using proportional (P), proportional-integral (PI), Inverse RLS-AVC, Inverse Neuro- AVC, Inverse RLS-AVC with gain and Inverse Neuro-AVC with gain control strategies are 9.840 dB, 6.840 dB, 9.380 dB, 8.590 dB, 17.240 dB and 5.770 dB, respectively