Exponential Fitting Technique Research Articles

Seasonal Swings and Graphical Tale of Growth and Productivity of Wheat in Gujarat State in Comparison with National Market (India)

This study comprehensively analyzes the growth and variability in the area, production, and productivity of wheat, a staple food grain in Gujarat state, India, for the period 2001-2022. Wheat holds significant importance in India's agricultural landscape, being one of the staple crops vital for food security. Gujarat, known for its diverse agro-climatic conditions, plays a pivotal role in contributing to the nation's wheat production. Understanding the seasonal dynamics, growth patterns, and productivity trends of wheat in Gujarat which is the broader national market provides valuable insights into the agricultural dynamics of the region. Utilizing exponential fitting and statistical regression techniques, the research presents compound growth rates for India and Gujarat, revealing significant positive correlations between wheat cultivation area, production, and productivity. Graphical representations depict trends and fluctuations, highlighting periods of deviation from trend lines. Instability analyses for India and Gujarat shed light on the consistency and fluctuations in wheat cultivation, production, and productivity. Additionally, the study explores seasonal variations in wheat prices across markets in Gujarat, providing insights into intra-year price cycles and coefficient variations. The findings contribute valuable insights into the dynamics of wheat agriculture in Gujarat, offering implications for policy and agricultural practices.

Approximate Solution of a Class of Highly Oscillatory Integral Equations Using an Exponential Fitting Collocation Method

This paper deals with the numerical solution of a class of highly oscillatory Volterra integral equations by collocation methods based on the exponential fitting technique. By reviewing the oscillatory structures of solutions of these problems, we construct an exponential fitting collocation method which is best tuned to capture the qualitative behaviour of the solution of these equations. We also investigate the convergence properties of the proposed collocation solution based on the interpolation remainder. Some numerical examples are provided which illustrate the efficiency and accuracy of the proposed method and confirm its superiority over the polynomial collocation methods.

A hybrid mixed finite element method for convection-diffusion-reaction equation with local exponential fitting technique

A new hybrid mixed finite element method is proposed for solving the convection-diffusion-reaction equation with local exponential fitting technique. In this method, the exponential fitting technique in [7] is used to discretize the convection-diffusion-reaction equation by introducing a new variable at element level; then the hybrid mixed discontinuous Galerkin finite element method is used to approximate the discretized problem. The convergence of the proposed method is analyzed, and the corresponding a priori error estimate is derived. The numerical results are presented to confirm our theoretical analysis.

Photoluminescence and quantum yield behaviour of vanadium doped Zn2SiO4 phosphors facilely synthesized by sol-gel method

Vanadium-doped Zinc Silicate (Zn2SiO4) phosphors were synthesized through the sol-gel method. Structural, morphological and optical techniques were used to investigate the effects of vanadium incorporation on their structure, morphology, and optical behaviour. We reported that vanadium incorporation in Zn2SiO4 phosphors significantly modifies their crystallinity, morphology, and photoluminescence properties. When vanadium is added to Zn2SiO4 phosphors, the energy band gap (Eg) changes from 5.29 to 2.34 eV. Vanadium dopants generate imperfections in Zn2SiO4 phosphors, the leading cause of their emissions in visible regions and quantum yield. Quantum yield is estimated at 7.06% for Zn2−xVxSiO4 (x = 0.08) phosphor. The luminescence decay lifetime of the prominent emissions of vanadium-doped Zn2SiO4 was measured using a double exponential fitting technique, and the average lifetime is 11.7 ns.

Stability analysis of ef Gaussian direct quadrature methods for Volterra integral equations

The present paper analyzes the numerical stability of exponentially fitted Gaussian direct quadrature methods for Volterra integral equations. By applying the exponential fitting technique, the coefficients of these methods are specially tuned on the problem. The analysis of numerical stability of these methods with respect to the classical basic test equation is carried out, which takes into account the dependence of the methods both on the stepsize and on the problem parameters. Several stability regions are shown and some numerical experiments confirm theoretical expectations.

Exponentially fitted methods that preserve conservation laws

The exponential fitting technique uses information on the expected behaviour of the solution of ordinary and partial differential equations to define accurate and efficient numerical methods. In particular, exponentially fitted methods are very effective when applied to problems with oscillatory solutions. In these cases, they perform better than standard methods, particularly in the large time-step regime.In this paper we consider exponentially fitted Runge–Kutta methods and we give characterizations of those that preserve local conservation laws of linear and quadratic quantities.As a benchmark for the general theory we apply the symplectic midpoint and its exponentially fitted version to approximate breather wave solutions of partial differential equations arising as models in several fields, such as fluid dynamics and quantum physics. Numerical tests show that the exponentially fitted method performs better and its solution exactly satisfy discrete conservation laws of mass (or charge) and momentum.

Numerical simulation for time‐fractional nonlinear reaction–diffusion system on a uniform and nonuniform time stepping

In this article, two nonstandard high‐order schemes on a uniform and nonuniform time stepping combined with the multi‐parameter exponential fitting technique (MPEF) have been developed to solve the time‐fractional nonlinear reaction–diffusion system. The first method based on the MPEF combined with the 3‐weighted shifted‐Grünwald–Letnikov approximation with uniform time stepping, this scheme leads to a numerical solution that suffers from the singularity neart= 0. In order to frustrate this singularity, a nonstandard higher‐order L1‐approximation for a nonuniform time‐stepping scheme is developed. The developed scheme's convergence and unconditionally stability analysis have been verified. Numerical results effectively validate the theoretical aspects.

Time-fractional nonlinear Swift-Hohenberg equation: Analysis and numerical simulation

In this paper, a new scheme based on the exponential fitting technique is presented for solving the nonlinear time-fractional Swift-Hohenberg equation, where the first and second-order derivatives are replaced by Caputo fractional derivative. The exponential fitting technique depends on a parameter that led to getting high-order accuracy. The convergence and unconditional stability will be discussed using the Fourier method and the analysis is built on uniform and nonuniform time steps, to solve initial singularity by using the graded meshes. Applicability and theoretical results will be demonstrated and enhanced with numerical examples.

On the numerical stability of the exponentially fitted methods for first order IVPs

In the numerical solution of Initial Value Problems (IVPs) for differential systems, exponential fitting (EF) techniques are introduced to improve the accuracy behaviour of classical methods when some information on the solutions is known in advance. Typically, these EF methods are evaluated by computing their accuracy for some test problems and it is usual to assume that the stability behaviour is similar to the underlying classical methods. The aim of this paper it to show that for some standard methods the stability behaviour of their exponentially fitted versions may change strongly. Furthermore, this stability depends on the choice of the fitting space, that must be carefully selected in order to assess the quality of the integrators for the IVPs under consideration. In particular, we will show that for the usual fitting space 〈exp(ωx),exp(−ωx)〉 with ω∈R the size of the stability domain of the EF method can be much smaller than the one for the original method.

Thermal analysis for optimized selection of cooling techniques for SiC devices in high frequency switching applications

The inception of wide bandgap power semiconductors has paved way for the evolution of highly efficient power switches. A deep understanding about the thermal characteristics of the SiC power device is essential to utilize its advantages in terms of higher thermal and power handling capability. The primary objective of this paper is to establish an accurate thermal model by proposing and comparing different methodologies for thermal analysis of commercially available SiC device, which in turn will aid to select optimum cooling techniques for higher system efficiency. Exponential curve fitting technique was exploited to deduce the parameters for RC equivalent network model. A high frequency sinusoidal pulse width modulated SiC MOSFET inverter with a fixed load was considered for electro-thermal analysis using a unified software platform PLECS. The effect of switching frequency on heat sink volume was established through simulation studies with imbibed datasheet parameters of a commercial power device. Finally, a 3-D model of the commercially available SiC power MOSFET in SOT-227B package was built to analyse the device heat dissipation profile for an optimized cooling choice. The proposed modelling approach serves to provide a platform for the thermal optimization of power devices and can be extended to other semiconductor devices.

Exponentially fitted methods for solving time fractional nonlinear reaction–diffusion equation

In this article, we will develop a new numerical scheme with the second order in time and a class of fourth order or sixth order in space based on the exponential fitting techniques to approximate the nonlinear time fractional reaction–diffusion equation with fixed order and distributed order derivatives. These techniques depend on a parameter, which will be used to annihilate the error and increase the order of accuracy. The proposed methods are proved to be unconditionally stable and convergent by Fourier analysis. Also, the theoretical results and the effectiveness of the numerical scheme are confirmed by numerical test problems and a comparison with other methods is presented.

Exponentially fitted methods for solving two-dimensional time fractional damped Klein–Gordon equation with nonlinear source term

In this paper, we introduce a numerical study for the two-dimensional damped Klein–Gordon equation with time fractional and nonlinear source term. The proposed scheme is based on the exponential fitting technique and gives rise to a class of fourth order or sixth order in both spatial dimensions and with the second order in time. This technique depends on a free parameter, which will be used to annihilate the error and increase the order of accuracy. The proposed method's stability and convergence are analyzed by Fourier analysis. Also, the theoretical results and the effectiveness of the numerical scheme are confirmed by numerical tests.

Adapted numerical modelling of the Belousov–Zhabotinsky reaction

Adapted numerical schemes for the integration of differential equations generating periodic wavefronts have reported benefits in terms of accuracy and stability. This work is focused on differential equations modelling chemical phenomena which are characterized by an oscillatory dynamics. The adaptation is carried out through the exponential fitting technique, which is specially suitable to follow the apriori known qualitative behavior of the solution. In particular, we have merged this strategy with the information coming from existing theoretical studies and especially the observation of time series. Numerical tests will be provided to show the effectiveness of this problem-oriented approach.

Explicit Runge–Kutta Methods for Stiff Problems with a Gap in Their Eigenvalue Spectrum

In this paper we consider the numerical solution of stiff problems in which the eigenvalues are separated into two clusters, one containing the “stiff”, or fast, components and one containing the slow components, that is, there is a gap in their eigenvalue spectrum. By using exponential fitting techniques we develop a class of explicit Runge–Kutta methods, that we call stability fitted methods, for which the stability domain has two regions, one close to the origin and the other one fitting the large eigenvalues. We obtain the size of their stability regions as a function of the order and the fitting conditions. We also obtain conditions that the coefficients of these methods must satisfy to have a given stiff order for the Prothero–Robinson test equation. Finally, we construct an embedded pair of stability fitted methods of orders 2 and 1 and show its performance by means of several numerical experiments.

Thermal modeling of wide bandgap semiconductor devices for high frequency power converters

The emergence of wide bandgap semiconductors has led to development of new generation semiconductor switches that are highly efficient and scalable. To exploit the advantages of GaNFETs in power converters, in terms of reduction in the size of heat sinks and filters, a thorough understanding of the thermal behavior of the device is essential. This paper aims to establish a thermal model for wideband gap semiconductor GaNFETs commercially available, which will enable power electronic designers to obtain the thermal characteristics of the device more effectively. The model parameters is obtained from the manufacturer’s data sheet by adopting an exponential curve fitting technique and the thermal model is validated using PSPICE simulations. The model was developed based on the parametric equivalence that exists between the thermal and electrical components, such that it responds for transient thermal stresses. A suitable power profile has been generated to evaluate the GaNFET model under different power dissipation scenarios. The results were compared with a Silicon MOSFETs to further highlight the advantages of the GaN devices. The proposed modeling approach can be extended for other GaN devices and can provide a platform for the thermal study and heat sink optimization.

Parameter estimation in IMEX-trigonometrically fitted methods for the numerical solution of reaction–diffusion problems

In this paper, an adapted numerical scheme for reaction–diffusion problems generating periodic wavefronts is introduced. Adapted numerical methods for such evolutionary problems are specially tuned to follow prescribed qualitative behaviors of the solutions, making the numerical scheme more accurate and efficient as compared with traditional schemes already known in the literature. Adaptation through the so-called exponential fitting technique leads to methods whose coefficients depend on unknown parameters related to the dynamics and aimed to be numerically computed. Here we propose a strategy for a cheap and accurate estimation of such parameters, which consists essentially in minimizing the leading term of the local truncation error whose expression is provided in a rigorous accuracy analysis. In particular, the presented estimation technique has been applied to a numerical scheme based on combining an adapted finite difference discretization in space with an implicit–explicit time discretization. Numerical experiments confirming the effectiveness of the approach are also provided.

Determination of relaxation modulus of time-dependent materials using neural networks

Health monitoring systems for plastic based structures require the capability of real time tracking of changes in response to the time-dependent behavior of polymer based structures. The paper proposes artificial neural networks as a tool of solving inverse problem appearing within time-dependent material characterization, since the conventional methods are computationally demanding and cannot operate in the real time mode. Abilities of a Multilayer Perceptron (MLP) and a Radial Basis Function Neural Network (RBFN) to solve ill-posed inverse problems on an example of determination of a time-dependent relaxation modulus curve segment from constant strain rate tensile test data are investigated. The required modeling data composed of strain rate, tensile and related relaxation modulus were generated using existing closed-form solution. Several neural networks topologies were tested with respect to the structure of input data, and their performance was compared to an exponential fitting technique. Selected optimal topologies of MLP and RBFN were tested for generalization and robustness on noisy data; performance of all the modeling methods with respect to the number of data points in the input vector was analyzed as well. It was shown that MLP and RBFN are capable of solving inverse problems related to the determination of a time dependent relaxation modulus curve segment. Particular topologies demonstrate good generalization and robustness capabilities, where the topology of RBFN with data provided in parallel proved to be superior compared to other methods.

Revised exponentially fitted Runge–Kutta–Nyström methods

It is the purpose of this paper to revise the exponential fitting technique for the numerical solution of special second order ordinary differential equations (ODEs) y″=f(x,y), with oscillatory or periodic solutions, by Runge–Kutta–Nyström methods. Due to the multistage nature of these methods, the proposed technique takes into account the contribution to the error arising from the computation of the internal stages. The benefit on the accuracy of the overall numerical scheme is visible in the presented numerical evidence.

A new k-interval selection technique for fast atmospheric radiance calculation in remote sensing applications

A new technique is proposed to generate the k-interval parameters, including the number of k-intervals, the equivalent absorption coefficients, and the quadrature weights when using the correlated k-distribution method for the computation of spectrally integrated three-dimensional (3D) atmospheric radiance. The main difference between the proposed technique and the traditional exponential sum fitting technique is that only quadrature weights are involved in the optimization process. To avoid the ill-conditioned problem in the proposed technique, the absorption coefficients with high value are dealt with by the delta log( k) (Δlog( k)) technique instead of involving them in the fitting procedure. The performance of the proposed technique is illustrated by radiance calculation results of cloudless and cloudy atmosphere for three different band settings. Results show that there are less relative errors with the proposed optimization technique than with the Δlog( k) technique under the same number of k-intervals. However, as the absorption becomes stronger, the performance of the proposed technique gradually decreases to the Δlog( k) technique. The relative root-mean-square error (RMSE) of radiance for 3D cloudy atmosphere can be controlled in less than 2% when the number of k-intervals is less than 10 particularly for weak absorption band, the RMSEs are less than 1% with only 6 terms.

Exponentially fitted two-step hybrid methods for [formula omitted

It is the purpose of this paper to derive two-step hybrid methods for y″=f(x,y), with oscillatory or periodic solutions, specially tuned to the behaviour of the solution, through the usage of the exponential fitting technique. The construction of two-step exponentially fitted hybrid methods is shown and their properties are discussed. Some numerical experiments confirming the theoretical expectations are provided.