This article presents a finite element scheme with Newton's method for solving the time-fractional nonlinear diffusion equation. For time discretization, we use the fractional Crank-Nicolson scheme based on backward Euler convolution quadrature. We discuss the existence-uniqueness results for the fully discrete problem. A new discrete fractional Gronwall type inequality for the backward Euler convolution quadrature is established. A priori error estimate for the fully discrete problem in L 2 (Ω) norm is derived. Numerical results based on finite element scheme are provided to validate theoretical estimates on time-fractional nonlinear Fisher equation and Huxley equation. KEYWORDS discrete fractional Gronwall type inequality, error estimates, fractional Crank-Nicolson method, time-fractional diffusion equation 1 Present address Sudhakar Chaudhary,
This paper deals with two fractional Crank-Nicolson-Galerkin finite element schemes for coupled time-fractional nonlinear diffusion system. The first scheme is iterative and is based on Newton's method, while the other one is a linearized scheme. Existence-uniqueness results of the fully discrete solution for both schemes are discussed. In addition, a priori bounds and a priori error estimates are derived for proposed schemes using a new discrete fractional Grönwall-type inequality. Both the schemes yield O(Δt 2) accuracy in time and hence, superior to O(Δt 2−α) accurate L1 scheme existing in the literature. Moreover, three different numerical examples are provided to illustrate the theoretical estimates .
Traditionally permanent acoustic sensors leak detection techniques have been proven to be very effective in water distribution pipes. However, these methods need long distance deployment and proper position of sensors and cannot be implemented on underground pipelines. An inline-inspection acoustic device is developed which consists of acoustic sensors. The device will travel by the flow of water through the pipes which record all noise events and detect small leaks. However, it records all the noise events regarding background noises, but the time domain noisy acoustic signal cannot manifest complete features such as the leak flow rate which does not distinguish the leak signal and environmental disturbance. This paper presents an algorithm structure with the modularity of wavelet and neural network, which combines the capability of wavelet transform analyzing leakage signals and classification capability of artificial neural networks. This study validates that the time domain is not evident to the complete features regarding noisy leak signals and significance of selection of mother wavelet to extract the noise event features in water distribution pipes. The simulation consequences have shown that an appropriate mother wavelet has been selected and localized to extract the features of the signal with leak noise and background noise, and by neural network implementation, the method improves the classification performance of extracted features.
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