First Difference Research Articles

The MLS-based numerical manifold method for nonlinear transient heat conduction problems in functionally graded materials

In this study, we aim to numerically solve two-dimensional nonlinear transient heat conduction problems of functionally gradient materials (FGMs), where the governing equation is a quasilinear partial differential equation of second order. For this purpose, the weak form of the initial boundary value problem is first established. Then influence domains of nodes in the moving least squares (MLS) instead of finite elements are used as the mathematical patches to construct the mathematical cover of the numerical manifold method (NMM); while the shape functions of MLS-nodes as the weight functions subordinate to the mathematical cover, leading to the moving least squares based numerical manifold method (MLS-NMM). With the weak form, the spatial discretization is carried out using MLS-NMM, whereas temporal discretization follows the Euler backward difference. Finally, a series of numerical experiments are conducted concerning transient heat conduction problems of FGMs with heat source and heat convection, suggesting that the proposed MLS-NMM enjoys advantages of both MLS and NMM in solving nonlinear transient heat conduction of FGMs.

Electrochemical Hydrogen Compression: Modeling, Internal States Estimation and System Control

Hydrogen is a clean and flexible energy carrier that can be produced from diverse renewable energy sources (e.g., wind, solar and biomass) and used in a broad range of applications (e.g., transportation and power generation). Due to its high efficiency and carbon-free emission, hydrogen will play a key role in transition to Net Zero Emissions by 2050. However, for a successful hydrogen economy to develop, the compression and storage costs of hydrogen must be lowered to overcome its low volumetric energy density (i.e., 0.01079 MJ/L at STP). Currently， conventional mechanical compression accounts for the largest percentage of operating costs in hydrogen refueling stations [1].Electrochemical hydrogen compression (EHC) is an innovative non-mechanical technology that compresses hydrogen through application of voltage across membrane separators. Without the vulnerable moving parts of mechanical compressors, EHC is vibration-free, and the noise and possibility of compressor failure are reduced. Electrochemical compression is isothermal, in theory, and thus a higher efficiency is obtained than the adiabatic process of conventional mechanical compression. Consequently, EHC technology is attracting more and more attention as the hydrogen economy takes off.The proton conductivity of the proton exchange membrane (PEM) used in EHC is dependent on its water content and has a significant impact on the EHC performance. Unlike in PEM fuel cells, water is not a reaction product in EHC. Therefore, it is necessary to develop a control system to monitor the hydration degree of the membrane. Besides water management, thermal management is also important. Since high temperature can affect the stability of perfluoro-sulfonate polymers like Nafion®, their operating temperatures are typically below 353 K [2]. Reasonable optimization of operating conditions including current density, back pressure, gas flow, and inlet relative humidity, can effectively manage the temperature and moisture inside the EHC. However, the real-time internal states (e.g., temperature and water content) of EHC must be determined. Mathematical model of EHC with detailed geometries can be built to reveal the multiple coupling phenomena for mass/heat transfer, fluid flow and electrochemical reaction inside the EHC. Also, the EHC model can be used for system control and fault diagnosis.In this study, a high-fidelity dynamic model of EHC is developed with due consideration of two-dimensional distributed mass/heat transfer coupled with electrochemical kinetics. In the EHC model, the dynamics inside the anode and cathode gas channels are characterized by the discretized conservation equations of mass and energy balances, and a lumped parameter model is used to describe the transport and electrochemical reaction in the membrane electrode assembly (MEA). A numerical approach is implemented to solve the spatial derivatives of the gas channel model. Using appropriate boundary conditions, forward or backward differences are employed to realize finite-difference discretization. In the n-discretized finite elements model of EHC, the inlet and outlet flow of cathode and anode gas channels are considered as the boundary conditions. The reaction gas and water transport terms are considered as input disturbances for the discretized model, which can be determined based on the lumped MEA model.The EHC model can be used to detect local dehydration of PEM due to the unbalanced contribution of the electro-osmotic flow and the back diffusion of water across the membrane. The heterogeneities significantly affect the overall efficiency of an EHC. The physical parameters, such as the inlet relative humidity, discharge pressure, and PEM thickness, are optimized to enhance the overall efficiency of the system.[1] Parks, G., Boyd, R., Cornish, J., & Remick, R. (2014). Hydrogen station compression, storage, and dispensing technical status and costs: Systems integration (No. NREL/BK-6A10-58564). National Renewable Energy Lab. (NREL), Golden, CO (United States).[2] Sdanghi, G., Maranzana, G., Celzard, A., & Fierro, V. (2019). Review of the current technologies and performances of hydrogen compression for stationary and automotive applications. Renewable and Sustainable Energy Reviews, 102, 150-170.

Error analysis of fast L1 ADI finite difference/compact difference schemes for the fractional telegraph equation in three dimensions

This article proposes the fast L1 alternating direction implicit (ADI) finite difference and compact difference schemes to solve the fractional telegraph equation in three-dimensional space. The fully-discrete fast L1 ADI finite difference scheme can be established via the fast L1 formula for the approximation of mixed Caputo fractional derivatives and the central difference formula for the approximation of the spatial derivative term, then from which an ADI algorithm is designed to reduce three-dimensional problems to a series of one-dimensional problems. We add the corresponding compact operators in all directions of the space to get the fully-discrete L1 ADI compact difference scheme. Then the convergence in L2 and H1 norms of two ADI schemes is derived via energy method. Eventually, numerical experiments are carried out to verify the theoretical estimates.

A linearized finite difference scheme for time–space fractional nonlinear diffusion-wave equations with initial singularity

Abstract This paper presents a linearized finite difference scheme for solving a kind of time-space fractional nonlinear diffusion-wave equations with initial singularity, where the Caputo fractional derivative in time and the Riesz fractional derivative in space are involved. First, the considered problem is equivalently transformed into its partial integro-differential form. Then, the fully discrete scheme is constructed by using the Crank–Nicolson technique, the L1 approximation, and the convolution quadrature formula to deal with the temporal discretizations. Meanwhile, the classical central difference formula and the fractional central difference formula are applied to approximate the second-order derivative and the Riesz fractional derivative in space, respectively. Moreover, the stability and convergence of the proposed scheme are strictly proved by using the discrete energy method. Finally, some numerical experiments are presented to illustrate the theoretical results.

A higher‐order hybrid spline difference method on adaptive mesh for solving singularly perturbed parabolic reaction–diffusion problems with robin‐boundary conditions

AbstractWe propose a hybrid numerical scheme to discretize a class of singularly perturbed parabolic reaction–diffusion problems with robin‐boundary conditions on an equidistributed grid. The hybrid difference scheme is developed by using a modified backward difference scheme in time, a combination of the cubic spline and exponential spline difference scheme in space. The proposed scheme uses a cubic spline difference scheme for the discretization of robin‐boundary conditions. For the time discretization of the problem, we use the standard uniform mesh while a layer adapted equidistributed grid is generated for the spatial discretization. By equidistributing a curvature‐based monitor function, the spatial adaptive grid is able to capture the presence of parabolic boundary layers without using any prior information about the solution. Parameter uniform error estimates are derived to illustrate an optimal convergence of first‐order in time and second‐order in space for the proposed discretization. The accuracy of the proposed scheme is confirmed by the numerical experiments that underpin the theoretical analysis.

A new numerical formulation for the generalized time-fractional Benjamin Bona Mohany Burgers’ equation

Abstract In this article, we present a novel numerical formulation for the generalized time-fractional Benjamin Bona Mohany Burgers’ (BBMB) equation using Atangana Baleanu Caputo (ABC) derivative. First, we apply a linearization technique to deal with the generalized non-linear expression, and then the Crank–Nicolson finite difference formula is used in the temporal direction. A reliable numerical technique is applied to discretize the time-fractional ABC derivative, and the central difference formulae are used to approximate the derivatives in the spatial direction. The method is shown unconditionally stable and second-order convergent in both directions through the Fourier analysis. The numerical results of two test problems are analyzed to validate the theoretical results.

Newton-series-based observer–predictor control for disturbed input-delayed discrete-time systems

This paper deals with the problem of predicting the future state of discrete-time input-delayed systems in the presence of unknown disturbances that can affect both the state and the output equations of the plant. Since the disturbance is unknown, computing an exact prediction of the future plant states is not possible. To circumvent this problem, we propose using a high-order extended Luenberger-type observer for the plant states, disturbances, and their finite difference variables, combined with a new equation for computing the prediction based on Newton’s series from the calculus of finite differences. Detailed performance analysis is carried out to show that, under certain assumptions, both enhanced prediction and improved attenuation of the unknown disturbances are achieved. Linear matrix inequalities (LMIs) are employed for the observer design to minimize the prediction errors. A stabilization procedure based on an iterative design algorithm is also presented for the case where the plant is affected by time-varying uncertainties. Examples from the literature illustrate the advantages of the scheme.

Urdu translation and validation of the international consultation on Incontinence Questionnaire-Female Lower Urinary Tract Symptoms (ICIQ-FLUTS).

The International Consultation on Incontinence Questionnaire-Female Lower Urinary Tract Symptoms (ICIQ-FLUTS) is a tool for the evaluation of lower urinary tract symptoms and their impact on the quality of life. The purpose of this research was the translation of ICIQ-FLUTS into the Urdu language and to determine its validity and reliability. Two bilingual translators translated the ICIQ-FLUTS Questionnaire into the Urdu language. Three experts compared the Urdu ICIQ-FLUTS with the original version of ICIQ-FLUTS. Urdu version was translated back into English by two translators. The pre-final version was drafted after tracking the forward and backward translation differences. The final Urdu version was drafted after application to ten patients (pilot study). Final Urdu ICIQ-FLUTS was applied to 120 participants for validity and reliability. Urdu version was drafted after the consensus of the experts. The content validity index ranged between 0.81-0.93. The mean age of the participants was 45.2 ± 7.79 years. The test-retest reliability was analyzed through intraclass correlation coefficient (ICC) 0.972 and internal consistency by Cronbach's alpha 0.986. The concurrent validity was determined by correlating the ICIQ-FLUTS with ICIQ-UI SF (0.82) and UDI-6 (0.72). It was concluded that the Urdu version of ICIQ-FLUTS is an assessment questionnaire for the signs and symptoms of urinary incontinence. It has good content validity, concurrent validity, and test-retest reliability.

Extended FDTD Method for Coanalysis of Superconducting Passive Transmission Lines and Josephson Junction Drivers and Receivers

In rapid single-flux quantum circuits, the signal integrity problem of superconducting passive transmission lines (PTLs) becomes critical. In this article, an extended 3-D finite-difference time-domain (FDTD) method is proposed for coanalysis of superconducting PTLs and Josephson junction (JJ) drivers and receivers. The central difference FDTD scheme is used for PTLs, and the backward difference method is adopted for JJs, which are introduced to FDTD cells as lumped elements. Due to the nonlinearity of JJs, the Newton–Raphson method is introduced to solve the boundary condition of the FDTD method. Compared with the 1-D FDTD method of the transmission line model of PTLs, the proposed method performs full-wave analysis of the physical structure of PTLs. Therefore, the proposed 3-D FDTD method is theoretically more accurate than the conventional method by avoiding parameter extraction and circuit analysis.

Error Analysis of a PFEM Based on the Euler Semi-Implicit Scheme for the Unsteady MHD Equations

In this article, we mainly consider a first order penalty finite element method (PFEM) for the 2D/3D unsteady incompressible magnetohydrodynamic (MHD) equations. The penalty method applies a penalty term to relax the constraint “”, which allows us to transform the saddle point problem into two smaller problems to solve. The Euler semi-implicit scheme is based on a first order backward difference formula for time discretization and semi-implicit treatments for nonlinear terms. It is worth mentioning that the error estimates of the fully discrete PFEM are rigorously derived, which depend on the penalty parameter , the time-step size , and the mesh size h. Finally, two numerical tests show that our scheme is effective.

A fully explicit staggered algorithm for near‐field wave propagation of fluid‐saturated porous media based on u‐p dynamic formulation

AbstractNear‐field wave propagation of the fluid‐saturated porous media (FSPM) is an important issue in soil dynamics and geotechnical earthquake engineering. In the current study, a fully explicit staggered algorithm for the near‐field wave propagation analysis of the FSPM in the time domain is developed based on the u‐p dynamic formulation. The decoupling of dynamic formulation is implemented by the diagonalization of the mass matrix and pore fluid compressibility matrix. The central difference method and Newmark constant average acceleration method are used for the solution of the displacement and velocity of solid skeleton, and the backward difference method is used for the solution of pore fluid pressure in the time domain. The effectiveness and feasibility of the matrix diagonalization are verified by the comparison with the classical numerical algorithm. The proposed algorithm is also validated by the comparison of the numerical results with the corresponding analytical solution results. Combing with the transmitting artificial boundary condition, the proposed algorithm is then applied to investigate the seismic response of a FSPM free field, and its applicability to the typical near‐field wave propagation problems of the FSPM is indicated. The stability analysis of the proposed algorithm is finally conducted, and the corresponding stability criterion is presented. For the proposed algorithm, all the dynamic variables of the FSPM are solved in an iterative way and the coupling dynamic equations need not to be solved at each time step. The computational effort can be reduced considerably, and the computational efficiency can be improved remarkably.

A novel method for deinterleaving radar signals: First‐order difference curve based on sorted TOA difference sequence

Radar signal deinterleaving is used to separate interleaved pulse streams in the electronic support measure (ESM) systems. The histogram methods based on the difference in time of arrival (TOA) are more mature in engineering applications than other methods because of their more effortless implementation. In the increasingly complex electronic battlefield, pulse repetition interval (PRI) jitter, and pulse missing are inevitable in the interleaved pulse stream. The TOA differences of pulse trains with PRI jitter distributes in multiple adjacent PRI bins, which results in the significant reduction of the corresponding histogram value. Furthermore, the histogram values of real PRIs vary with pulse missing, which indicates that the histogram lacks robustness to pulse missing. First-order difference curve based on sorted TOA difference sequence, denoted as FDC-DTOA, is proposed in this paper to overcome the shortcomings mentioned above. In this method, multi-order TOA differences are calculated and then sorted in ascending order. Then, FDC-DTOA is obtained after a first-order backward difference operation. Finally, an adaptive threshold is applied to FDC-DTOA to extract candidate PRIs. The simulation results show that the proposed method has remarkable robustness to pulse missing and excellent performance in deinterleaving the pulse stream with PRI jitter.

ANDERSON’S ∇- INTEGRAL INEQUALITY

Basically, time scale calculus is the theory of unification of traditional calculus with that calculus of difference i.e. discrete calculus. Time Scale Calculus is a field of discussion in the area of traditional analysis of mathematics. It focuses on the dynamic system which has a lot of applications in various fields of life. Calculus of time scales is a valuable field due to numerous applications in covid-19 disease cases. Notably, Time scale calculus has a long relation with mathematical inequalities that can be discussed with fractional calculus. The Anderson Integral Inequality, which provides a lower constraint for the integration of convex mapping in the form of the averages of each constituent, is described in this research paper on ∇- time-scale calculus. On ∇-time scale we formulated Anderson’s integral inequality as given below: if φ_j (j=1,….,α) accomplish some appropriate cases.

Radial basis function partition of unity procedure combined with the reduced-order method for solving Zakharov–Rubenchik equations

A meshless radial basis function based on partition of unity (RBF-PU) method is proposed to solve Zakharov–Rubenchik equations. In this local method, the domain is split into overlapping patches forming a covering of it and also, it provides accurate results for PDEs. Time discretization is performed using a second-order implicit explicit backward difference method (IMEX-BDF2). Although the proper orthogonal decomposition (POD) is applied to reduce the dimension of the governing model, the computational complexity of the reduced model for nonlinear terms still depends on the number of variables of the full model. To overcome this subject, we employ the discrete empirical interpolation method (DEIM). Two problems with different situations are solved by the proposed method and the comparison of numerical findings with the conservative compact difference scheme and RBF-FD method shows that the presented method provides accurate results at a low computing cost.

Convergence analysis of a fast second‐order time‐stepping numerical method for two‐dimensional nonlinear time–space fractional Schrödinger equation

AbstractIn this article, we consider the two‐dimensional nonlinear time–space fractional Schrödinger equation with space described by the fractional Laplacian. A second‐order fractional backward difference formula in the temporal direction while Fourier spectral method in the spatial direction is proposed to solve the model numerically. In the numerical implementation, a fast method is applied based on a globally uniform approximation of the trapezoidal rule for the integral on the real line to decrease the memory requirement and computational cost. By using the generalized discrete Gronwall inequality developed by Dixon and McKee and the temporal–spatial error splitting argument, the convergence of the fast time‐stepping numerical method is also proved in a simple manner without imposing the Courant‐Friedrichs‐Lewy (CFL) condition. Finally, some numerical results are provided to support the theoretical analysis.

A domain decomposition method of Schwarz waveform relaxation type for singularly perturbed nonlinear parabolic problems

We introduce a domain decomposition method of discrete Schwarz waveform relaxation (DSWR) type for a singularly perturbed nonlinear parabolic problem. The method utilizes Shishkin transition parameter for a space–time decomposition of the computational domain. In each subdomain, the problem is discretized using the central differencing and backward difference schemes on a uniform mesh in space and time directions, respectively. Further, the exchange of information between the subdomains is done through the Dirichlet data that leads to optimal convergence. We analyse the convergence of the developed method and show that the method converges very fast for small perturbation parameter and provides uniformly convergent approximations to the solution of the nonlinear problem. Finally, with some numerical experiments, we illustrate our theoretical results.

A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions. An alternative approach

A numerical technique is devised to solve a problem of the steady laminar fluid motion in a straight plane hard-walled duct with two local axisymmetric rectangular constrictions. It uses the stream function, the vorticity and the pressure as the basic variables, has the second order of accuracy in the spatial and the first order of accuracy in the temporal coordinates, provides high stability of a solution, and needs significantly less computational time to obtain a result compared to appropriate techniques available in a scientific literature. The technique consists in: a) introducing the stream function and the vorticity, and subsequent transiting from the non-dimensional governing relations for the fluid velocity and the pressure to the corresponding non-dimensional relations for the stream function, the vorticity and the pressure; b) deriving their discrete counterparts in the nodes of the chosen space-time computational grid; c) integrating the systems of linear algebraic equations obtained after making the discretization. The discretization is based on applying appropriate differencing schemes to the terms of the equations for the basic variables. These are the two-point temporal onward difference for the unsteady term of the vorticity equation, as well as the two-point backward differences (for its convective term) and the five-point approximations (for its diffusive term and for the Poisson’s equations for the stream function and the pressure) in the axial and cross-flow coordinates. As for the velocity components, the appropriate central differences are applied to discretize their expressions. The above-mentioned systems of linear algebraic equations for the stream function and the pressure are integrated by the iterative successive over-relaxation method. The algebraic relation for the vorticity does not need application of any method to be solved, because it is a computational scheme to find this quantity based on the known magnitudes computed at the previous instant of time.

Maximal regularity of backward difference time discretization for evolving surface PDEs and its application to nonlinear problems

Abstract Maximal parabolic $L^p$-regularity of linear parabolic equations on an evolving surface is shown by pulling back the problem to the initial surface and studying the maximal $L^p$-regularity on a fixed surface. By freezing the coefficients in the parabolic equations at a fixed time and utilizing a perturbation argument around the freezed time, it is shown that backward difference time discretizations of linear parabolic equations on an evolving surface along characteristic trajectories can preserve maximal $L^p$-regularity in the discrete setting. The result is applied to prove the stability and convergence of time discretizations of nonlinear parabolic equations on an evolving surface, with linearly implicit backward differentiation formulae characteristic trajectories of the surface, for general locally Lipschitz nonlinearities. The discrete maximal $L^p$-regularity is used to prove the boundedness and stability of numerical solutions in the $L^\infty (0,T;W^{1,\infty })$ norm, which is used to bound the nonlinear terms in the stability analysis. Optimal-order error estimates of time discretizations in the $L^\infty (0,T;W^{1,\infty })$ norm is obtained by combining the stability analysis with the consistency estimates.

Sliding mode control of discrete-time interval type-2 fuzzy Markov jump systems with the preview target signal

This paper studies the problem of output feedback sliding mode control for uncertain discrete-time interval type-2 (IT2) fuzzy Markov jump systems (MJSs) with the preview target signal. An augmented error system (AES) including the known preview target signal is derived based on the backward difference method. A sliding mode output controller is designed for the AES, which integrates the known future information. The designed controller guarantees the discrete-time reach condition and maintains the states in a neighborhood region of the predesigned sliding surface. The stability criteria of the sliding motion on a linear sliding surface are given for the considered systems. Finally, two practical examples show that the output of the original system can track the target signal better under the designed preview controller.

The finite difference method for the fourth‐order partial integro‐differential equations with the multi‐term weakly singular kernel

This paper is concerned with an efficient numerical method for a class of fourth‐order partial integro‐differential equations (PIDEs) with weakly singular kernel. Due to the presence of the weakly singular kernel, the exact solution has singularity near the initial time . The proposed method is constructed on the graded meshes, which can achieve the second‐order convergence in time for weakly singular solutions. The product integral rule of the Riemann–Liouville fractional integral with the generalized time‐stepping is used for the time discretization, and the standard central difference formula is used for the space discretization. The stability and convergence of the method are proved, and the optimal error estimates in the discrete ‐norm and ‐norm are obtained. Numerical results show the effectiveness of the proposed method. Further, by the extrapolation method, we improve the convergence order in space and time to four order, respectively.