Spline Collocation Approach Research Articles

A study on unsteady flow through porous media : A cubic spline collocation approach

An analysis carried out to study unsteady flow through porous media. The non-linear equation governing the flow problem is transformed into linear differential equation using quasi linearization technique and solved by applying cubic spline collocation method. The obtained numerical results are compared with finite difference method to show the effectiveness and accuracy of the method.

(Invited) Non Quasi Static Modeling of Printed Organic Thin Film Transistors in Large Signal and Small Signal Operation

As one of the main expected advantages of organic electronics is the low cost, printing technologies are actively investigated for the fabrication of OTFTs.Techniques such as ink-jet or gravure printing frequently results in organic semiconductor (OSC) layers with low crystallinity and relatively low carrier mobility. Moreover, the minimal feature sizes attainable are often in the range of the tens of microns. Hence, the resulting carrier transit time (τ) in the channel of OTFTs can be of the order of tenth of milliseconds with associated cut off frequencies in AC regime in the range of few kilohertz [1].Due to the large tolerances associated to printing processes, the layout of printed OTFTs often includes semiconductor regions whose dimensions are chosen prudently larger than the spacing existing between the source and drain contacts. The presence of the parasitic regions, that is negligible in DC operation, must be accounted in order to accurately reproduce the dynamic operation of these devices.Due to the relatively high value of τ, it is very likely that in practical applications organic circuits are operated with signals varying on a time scale of the same order or shorter than τ. Hence, the carriers in the device don’t have enough time to reach the distribution associated to stationary conditions and the OTFTs enter in non quasi static (NQS) regime of operation. The device response to large variation of its polarization (large signal, LS) becomes dependent on the delay times associated with the high value of τ. Moreover, in NQS regime the device effective admittance in small signal (SS) AC operation is found to be dependent upon the small signal frequency [2][3].Hence, to successfully design electrical circuits employing printed OTFTs, it is necessary to develop compact models that are able to precisely reproduce not only the DC operation but also the LS and SS AC behavior of these devices.In this work, a large signal non quasi static (LS-NQS) compact model has been developed: the model is based on the current continuity equation that, in the case of a MOSFET transistor channel, can be cast as a partial differential equation (PDE) governing the carrier density distribution during time, Q(x,t), in a distributed RC transmission line. In order to solve this PDE, the space variable x is discretized by using a spline collocation approach: the PDE is transformed in a set of ordinary differential equations (ODE) in the time variable t, that can be solved numerically in a “SPICE-like” circuit simulation environment [4], [5].The spline collocation approach allows to choose the number of splines (N) in which the device channels and the parasitic regions are split: by choosing an appropriate value for N it is possible to find the best compromise between computation accuracy and performance.NQS models have been already investigated and developed for crystalline silicon MOSFETs. The model presented in this work differs from the crystalline silicon counterparts in two major aspects:1 – the equations accounting for the charge and current densities are based upon the variable range hopping (VRH) theory that is well suited for amorphous organic semiconductors[6].2 – the model takes into account the presence of parasitic regions by imposing appropriate boundary conditions on their extremities.The LS-NQS model has been implemented in Verilog-A and validated by simulating the electrical behavior of multi-fingered OTFTs, with staggered top-gate configuration, fabricated on PEN substrates by means of a fully printed process. The devices were characterized and simulated in DC, and SS AC operation. A common source amplifier circuit has been realized and measured in LS operation. As can be seen from fig. 1, the model reproduce perfectly the device behavior in LS operation. Moreover, the use of the model has permitted to gain several physical insights concerning the device operation.[1] J. S. Chang et al., IEEE J. Emerg. Sel. Top. Circuits Syst., vol. 7, no. 1, pp. 7–26, Mar. 2017, doi: 10.1109/JETCAS.2017.2673863.[2] A. Valletta et al., IEEE Trans. Electron Devices, vol. 61, no. 12, pp. 4120–4127, Dec. 2014, doi: 10.1109/TED.2014.2364451.[3] J. Leise et al., IEEE Trans. Electron Devices, vol. 67, no. 11, pp. 4672–4676, Nov. 2020, doi: 10.1109/TED.2020.3018094[4] A. Valletta et al. in 2016 46th European Solid-State Device Research Conference (ESSDERC), Lausanne, Switzerland, Sep. 2016, pp. 460–463, doi: 10.1109/ESSDERC.2016.7599685.[5] A. Valletta et al. in 2017 European Conference on Circuit Theory and Design (ECCTD), Catania, Sep. 2017, pp. 1–4, doi: 10.1109/ECCTD.2017.8093225.[6] F. Torricelli et al., IEEE Trans. Electron Devices, vol. 56, no. 1, pp. 20–30, Jan. 2009, doi: 10.1109/TED.2008.2007717. Figure 1

Spline collocation approach to study Brachistochrone problem

In this paper authors discussed a problem of quickest descent, the Brachistochrone curve. Spline collocation method is used to solve the non-linear boundary value problem. The numerical results obtained are compared with the transformation method to show effectiveness and accuracy of this method.

Spline collocation approach to study two-dimensional and axisymmetric unsteady flow

Spline collocation approach to study two-dimensional and axisymmetric unsteady flow

A spline collocation approach for a business cycle model with small gestation period in production

A B-spline collocation approach is presented for the numerical solution of a business cycle model with a small gestation period in production. The differential equation that models the problem comprises of a nonlinearity that can arise either from the dependence of investment on profits, a Kaldorian-type model, or on capital stocks. The efficiency of the scheme is tested on a number of special cases; it is shown that the nonlinearity leads to a conditionally stable cycle. The rate of convergence is discussed theoretically and verified numerically to be of fourth-order.

NUMERICAL SOLUTION OF A CLASS OF NONLINEAR SYSTEM OF SECOND-ORDER BOUNDARY-VALUE PROBLEMS: A FOURTH-ORDER CUBIC SPLINE APPROACH

A cubic B-spline collocation approach is described and presented for the numerical solution of an extended system of linear and nonlinear second-order boundary-value problems. The system, whether regular or singularly perturbed, is tackled using a spline collocation approach constructed over uniform or non-uniform meshes. The rate of convergence is discussed theoretically and verified numerically to be of fourth-order. The efficiency and applicability of the technique are demonstrated by applying the scheme to a number of linear and nonlinear examples. The numerical solutions are contrasted with both analytical and other existing numerical solutions that exist in the literature. The numerical results demonstrate that this method is superior as it yields more accurate solutions.

An efficient collocation algorithm for multidimensional wave type equations with nonlocal conservation conditions

In this paper, we derive and analyze an efficient spectral collocation algorithm to solve numerically some wave equations subject to initial-boundary nonlocal conservation conditions in one and two space dimensions. The Legendre pseudospectral approximation is investigated for spatial approximation of the wave equations. The Legendre–Gauss–Lobatto quadrature rule is established to treat the nonlocal conservation conditions, and then the problem with its nonlocal conservation conditions are reduced to a system of ODEs in time. As a theoretical result, we study the convergence of the solution for the one-dimensional case. In addition, the proposed method is extended successfully to the two-dimensional case. Several numerical examples with comparisons are given. The computational results indicate that the proposed method is more accurate than finite difference method, the method of lines and spline collocation approach.

Coupled ABC and Spline Collocation Approach for a Classof Nonlinear Boundary Value Problems over Semi-InfiniteDomains

Coupled ABC and Spline Collocation Approach for a Classof Nonlinear Boundary Value Problems over Semi-InfiniteDomains

Two-Point Boundary-Value Problems over a Semi-Infinite Domain: A Coupled ABC and Spline Collocation Approach

The ultimate purpose of this article is to introduce and describe a combined approach, based on asymptotic boundary conditions (ABCs) and a fourth order cubic B-spline collocation, for the numerical solution of a general class of two-point linear boundary-value problems (BVPs) over a semi-infinite interval that arises in various engineering applications. The scheme will be extended and then implemented to handle a system of BVPs. The idea of the proposed strategy is to first reduce the condition at infinity to an asymptotic boundary condition that approaches the specified value at infinity over a large finite interval. Then, the problem complimented with the resulting ABC is solved using a fourth–order spline collocation approach constructed over uniform meshes. The scheme is numerically verified to have a fourth order rate of convergence. The work is illustrated by considering a number of test examples that confirm the accuracy, efficient treatment of the boundary condition at infinity, and applicability of the approach. The computational results show that the scheme is reliable, converges fast, and compares very well with the existing analytic solutions.

A shifted Jacobi collocation algorithm for wave type equations with non-local conservation conditions

Abstract In this paper, we propose an efficient spectral collocation algorithm to solve numerically wave type equations subject to initial, boundary and non-local conservation conditions. The shifted Jacobi pseudospectral approximation is investigated for the discretization of the spatial variable of such equations. It possesses spectral accuracy in the spatial variable. The shifted Jacobi-Gauss-Lobatto (SJ-GL) quadrature rule is established for treating the non-local conservation conditions, and then the problem with its initial and non-local boundary conditions are reduced to a system of second-order ordinary differential equations in temporal variable. This system is solved by two-stage forth-order A-stable implicit RK scheme. Five numerical examples with comparisons are given. The computational results demonstrate that the proposed algorithm is more accurate than finite difference method, method of lines and spline collocation approach

The boundary layer problem: A fourth-order adaptive collocation approach

A finite element approach, based on the cubic B-spline collocation, is presented for the numerical solution of a class of singularly perturbed two-point boundary value problems that possesses a boundary layer at one or two end points. Due to the existence of a layer, the problem is handled using an adaptive spline collocation approach constructed over a non-uniform Shishkin-like mesh, defined via a carefully selected generating function. To tackle the case of nonlinearity, if it exists, an iterative scheme arising from Newton’s method is employed.The rate of convergence is verified to be of fourth-order and is calculated using the double-mesh principle. The efficiency and applicability of the method are demonstrated by applying it to a number of linear and nonlinear examples. The numerical solutions are compared with both analytical and other existing numerical solutions in the literature. The numerical results confirm that this method is superior when contrasted with other accessible approaches and yields more accurate solutions.

A spline collocation approach for a generalized parabolic problem subject to non-classical conditions

A numerical method based on finite differences and spline collocation is proposed for solving a generalized parabolic problem subject to non-classical conditions. It is shown numerically that this scheme has second-order convergence. The stability conditions relating the time step size to mesh sizes were examined through numerical examples. It turns out that the stability of the scheme in certain cases is subjected to a stability condition and depends on the nature of the problem. A number of specific test problems are solved to assess the efficiency and performance of the method. The numerical results obtained indicate that the approach is viable and yields accurate solutions when compared with the existing closed-form solutions and generates results consistent with recent numerical approaches.

Troesch’s problem: A B-spline collocation approach

A finite-element approach, based on cubic B-spline collocation, is presented for the numerical solution of Troesch’s problem. The method is used on both a uniform mesh and a piecewise-uniform Shishkin mesh, depending on the magnitude of the eigenvalues. This is due to the existence of a boundary layer at the right endpoint of the domain for relatively large eigenvalues. The problem is also solved using an adaptive spline collocation approach over a non-uniform mesh via exploiting an iterative scheme arising from Newton’s method. The convergence analysis is discussed and is shown to depend on the eigenvalues; in particular, the rate of convergence is calculated using the double-mesh principle. To demonstrate the efficiency of the method, a number of special cases are considered. The numerical solutions are compared with both the analytical solutions and other existing numerical solutions in the literature. It is observed that the results obtained by this method are quite satisfactory and accurate, and the method is applicable for a wide range of cases when contrasted with other available solutions.

A spline collocation approach for a generalized wave equation subject to non-local conservation condition

In the present paper, a collocation finite element approach based on cubic splines is presented for the numerical solution of a generalized wave equation subject to non-local conservation condition. The efficiency, accuracy and stability of the method are assessed by applying it to a number of test problems. The results are compared with the existing closed-form solutions; the scheme demonstrates that the numerical outcomes are reliable and quite accurate when contrasted with the analytical solutions and an existing numerical method.

A spline collocation approach for the numerical solution of a generalized nonlinear Klein–Gordon equation

In this paper, a finite element collocation approach using cubic B-splines is employed for the numerical solution of a generalized form of the nonlinear Klein–Gordon equation. The efficiency of the method is tested on a number of examples that represent special cases of the extended equation including the sine-Gordon equation. The numerical results are compared with existing numerical and analytic solutions and the outcomes confirm that the scheme yields accurate and reliable results even when few nodes are used at the time levels.

Spline collocation approach to boundary value problems

AbstractA well‐known Stokes problem is discussed by a cubic spline collocation method. Two consecutive cubic splines are obtained for the problem. The results by this method are compared with those of an orthogonal collocation method. The selection of the length of the subintervals of the range of the boundary value problem is also justified. The results obtained by these two methods are compared with the analytic solution. The methods involve simple algebra, and hence the calculations do not require the help of a computer. Necessary error analysis has been carried out.