Abstract: This paper presents a novel two-dimensional real-time modeling approach for a proton exchange membrane fuel cell (PEMFC) based on a tridiagonal matrix algorithm (Thomas algorithm). The Thomas algorithm consists of a forward elimination and a backward substitution, its arithmetic complexity of computations being much lower than the Gaussian elimination. In order to use this advanced numerical solver, the differential equations of reactant gas convection and diffusion phenomena in serpentine channels are transformed into a tridiagonal equations system. In addition, a three-level bisection algorithm has been developed to solve spatial physical quantities distribution for electrochemical domain. The real-time computing methods developed in this paper are then implemented in C language for a fast execution time in a real-time processor. The proposed real-time model is experimentally validated using a 1.2 kW Ballard NEXA fuel cell system, and its practical feasibilities in advanced real-time control for PEMFC systems have been experimentally demonstrated in an RT-LAB real-time simulator.

Abstract: A mathematical model is developed for solar drying of green peas (Botanical name: Pisum Sativum). The problem is solved assuming the shape of the green peas is spherical. The governing transient mass transfer equation is discretized into finite difference scheme. The time marching is performed by implicit scheme. The governing equations and boundary conditions are non-dimensionalized to get generic results. The product in the chamber is in contact with air which is heated by solar energy, so the boundary conditions of third kind (convective boundary conditions) are considered. By space and time discretization a set of algebraic equations are generated and these algebraic equations are solved by tridiagonal matrix algorithm. A computer code is developed in MATLAB in order to compute the transient moisture content distribution inside the product. Center point, boundary and mean moisture of green peas are estimated at different temperatures and drying time. Present numerical result is compared with experimental result from literature and it was found that there is a good agreement of results. The drying time is predicted for how quickly the mean moisture of green peas is reached to 50, 40, 30, 20 and 10% of its initial moisture corresponding to different temperatures.

Abstract: This paper is concerned with the numerical solution of the singularly perturbed differential-difference equations with small shifts called delay and advanced parameters. A fourth order finite difference method with a fitting factor is proposed for the solution of the singularly perturbed differential-difference equations with mixed shifts. The delay and advanced shifts are managed by Taylor series and an asymptotically equivalent singularly perturbed two-point boundary value problem is obtained. A fitting factor is introduced in the fourth order finite difference scheme for the problem which takes care of the small values of the perturbation parameter. This fitting factor is obtained from the asymptotic solution of singular perturbations. Thomas algorithm is used to solve the discrete system of the difference scheme. Convergence of the proposed method is analyzed. Maximum absolute errors in comparison with the several numerical experiments aretabulated to illustrate the proposed method.

Abstract: A real time empirical mode decomposition (EMD) algorithm based ultrasonic imaging system has been developed for non-destructive testing (NDT) applications. It is difficult to implement the EMD based signal processing algorithm in real time because it is totally a data-driven process which comprises numerous sifting operations. In this paper, the EMD algorithm has been implemented in the visual software environment. The EMD implementation encompasses two types of interpolation methods: piecewise linear interpolation (PLI) and cubic spline interpolation (CSI). The cubic spline tridiagonal matrix has been solved by using the Thomas algorithm for real time processing. The total time complexity functions for both the implemented PLI and CSI based EMD methods have been computed. For the signal filtering, the partial reconstruction algorithm has been adopted. The baseline correction and noise filtering applications have been presented using an EMD based visual software. The real time practicability and the efficiency of this method have been validated through ultrasonic NDT experimentation for improvement in the time domain resolution of the ultrasonic A-scan raw data. The practical results show that in the noisy environment, it is possible to enhance the signal-to-noise ratio for the visualization and identification of ultrasonic pulse-echo signals in real time.

Abstract: In this work, a fourth-order in space and second-order in time compact scheme, a sixth-order in space and second-order in time compact scheme and two linearized compact schemes are proposed for the (1+1)-dimensional nonlinear Dirac equation. The iterative algorithm is used to compute the nonlinear algebraic system and the Thomas algorithm in the matrix form is adopted to enhance the computational efficiency. It is proved that all of the schemes are unconditionally stable in the linear sense. Numerical experiments are given to test the accuracy order of the presented schemes, record the error history for all of the schemes with respect to t, discuss the conservation laws of discrete charge and energy from the numerical point of view, study the stability of the solitary waves by adding a small random perturbation to the initial data, and simulate the collision of two and three solitary waves.

Abstract: In this paper, we establish sufficient conditions for the existence, uniqueness and numerical solution for a parabolic integrodifferential equation with the second kind integral condition. The existence, uniqueness of a strong solution for the linear problem based on a priori estimate “energy inequality” and transformation of the linear problem to linear first-order ordinary differential equation with second member. Then by using a priori estimate and applying an iterative process based on results obtained for the linear problem, we prove the existence, uniqueness of the weak generalized solution of the integrodifferential prob- lem. Also we have developed an efficient numerical scheme, which uses temporary problems with standard boundary conditions. A suitable combination of the auxiliary solutions defines an approximate solution to the original nonlocal problem, the algebraic matrices obtained after the full discretization are tridiagonal, then the solution is obtained by using the Thomas algorithm. Some numerical results are reported to show the efficiency and accuracy of the scheme.

Abstract: In this paper, a new numerical integration method on a uniform mesh is presented for the solution of singularly perturbed two-point boundary value problems having boundary layer at one end (left or right) point. The methods of Exact and Trapezoidal rule of integration with finite difference approximation of first derivatives are used to obtain a three-term recurrence relationship . The obtained tridiagonal system of equations is then solved using Thomas algorithm. Also, the stability and convergence of the proposed scheme are established. Several model example problems are solved using the proposed method. The results are presented in terms of maximum absolute errors which demonstrate the accuracy and efficiency of the method. It is observed that the proposed method is capable of producing highly accurate results with minimal computational effort for a fixed value of step size h, when perturbation parameter tends to zero.

Abstract: Development in surface modifications including texturing and boundary slip has shown promising outcomes in enhancing the hydrodynamic lubrication performance. In this work, the modified Reynolds equation was considered in order to evaluate the tribocharacteristics of partially textured contact with boundary slip. Finite volume method coupled with tridiagonal matrix algorithm was used to solve non-linear Reynolds theory. For maximizing the load support, the optimization procedure was carried out using the exact optimization method. Results showed encouraging improvements in load support behaviour by shifting the multiple-texture to exit zone of the contact. It was also confirmed that the improvement of the load support of around 300% using the optimized textured lubricated contact could be achieved.

Abstract: Block tridiagonal matrices arise in applied mathematics, physics, and signal processing. Many applications require knowledge of eigenvalues and eigenvectors of block tridiagonal matrices, which can be prohibitively expensive for large matrix sizes. In this paper, we address the problem of the eigendecomposition of block tridiagonal matrices by studying a connection between their eigenvalues and zeros of appropriate matrix polynomials. We use this connection with matrix polynomials to derive a closed-form expression for the eigenvectors of block tridiagonal matrices, which eliminates the need for their direct calculation and can lead to a faster calculation of eigenvalues. We also demonstrate with an example that our work can lead to fast algorithms for the eigenvector expansion for block tridiagonal matrices.

Abstract: This paper developed a method called "modified exponential cubic B-Spline differential quadrature (mExp-DQM) for space discretization together with a time integration algorithm" for the numerical computation of hyperbolic telegraph equation in $(2+1)$ dimension. The mExp-DQM is a new differential quadrature method based on modified exponential cubic B-splines as basis which reduces the problem into an amenable system of ordinary differential equations. The resulting system is solved using a time integration algorithm. The stability of the method is also studied by computing the eigenvalues of the coefficients matrices, it is found that the scheme is conditionally stable. The accuracy of the method is illustrated by computing the error between analytical solutions and numerical solutions is measured by using $L_2$ and $L_{\infty}$ error norms for each problem. A comparison of mExp-DQM solutions with the results of the other numerical methods has been carried out for various space sizes and time step sizes.