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Mathematical models of various natural processes are described by differential equations, systems of partial differential equations and integral equations. In most cases, the exact solution to such problems cannot be determined; therefore, one has to use grid methods to calculate an approximate solution using high-performance computing systems. These methods include the finite element method, the finite difference method, the finite volume method and combined methods. In this Special Issue, we bring to your attention works on theoretical studies of grid methods for approximation, stability and convergence, as well as the results of numerical experiments confirming the effectiveness of the developed methods. Of particular interest are new methods for solving boundary value problems with singularities, the complex geometry of the domain boundary and nonlinear equations. A part of the articles is devoted to the analysis of numerical methods developed for calculating mathematical models in various fields of applied science and engineering applications. As a rule, the ideas of symmetry are present in the design schemes and make the process harmonious and efficient.

high-order methods --- Brinkman penalization --- discontinuous Galerkin methods --- embedded geometry --- high-order boundary --- IMEX Runge–Kutta methods --- boundary value problems with degeneration of the solution on entire boundary of the domain --- the method of finite elements --- special graded mesh --- multigrid methods --- Hermitian/skew-Hermitian splitting method --- skew-Hermitian triangular splitting method --- strongly non-Hermitian matrix --- lie symmetries --- invariantized difference scheme --- numerical solutions --- finite integration method --- shifted Chebyshev polynomial --- direct and inverse problems --- Volterra integro-differential equation --- Tikhonov regularization method --- quartic spline --- triangulation --- scattered data --- continuity --- surface reconstruction --- positivity-preserving --- interpolation --- jaw crusher --- symmetrical laser cladding path --- FEPG --- wear

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Short-term load forecasting (STLF) plays a key role in the formulation of economic, reliable, and secure operating strategies (planning, scheduling, maintenance, and control processes, among others) for a power system and will be significant in the future. However, there is still much to do in these research areas. The deployment of enabling technologies (e.g., smart meters) has made high-granularity data available for many customer segments and to approach many issues, for instance, to make forecasting tasks feasible at several demand aggregation levels. The first challenge is the improvement of STLF models and their performance at new aggregation levels. Moreover, the mix of renewables in the power system, and the necessity to include more flexibility through demand response initiatives have introduced greater uncertainties, which means new challenges for STLF in a more dynamic power system in the 2030–50 horizon. Many techniques have been proposed and applied for STLF, including traditional statistical models and AI techniques. Besides, distribution planning needs, as well as grid modernization, have initiated the development of hierarchical load forecasting. Analogously, the need to face new sources of uncertainty in the power system is giving more importance to probabilistic load forecasting. This Special Issue deals with both fundamental research and practical application research on STLF methodologies to face the challenges of a more distributed and customer-centered power system.

History of engineering & technology --- short-term load forecasting --- demand-side management --- pattern similarity --- hierarchical short-term load forecasting --- feature selection --- weather station selection --- load forecasting --- special days --- regressive models --- electric load forecasting --- data preprocessing technique --- multiobjective optimization algorithm --- combined model --- Nordic electricity market --- electricity demand --- component estimation method --- univariate and multivariate time series analysis --- modeling and forecasting --- deep learning --- wavenet --- long short-term memory --- demand response --- hybrid energy system --- data augmentation --- convolution neural network --- residential load forecasting --- forecasting --- time series --- cubic splines --- real-time electricity load --- seasonal patterns --- Load forecasting --- VSTLF --- bus load forecasting --- DBN --- PSR --- distributed energy resources --- prosumers --- building electric energy consumption forecasting --- cold-start problem --- transfer learning --- multivariate random forests --- random forest --- electricity consumption --- lasso --- Tikhonov regularization --- load metering --- preliminary load --- short term load forecasting --- performance criteria --- power systems --- cost analysis --- day ahead --- feature extraction --- deep residual neural network --- multiple sources --- electricity

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• Reference Manager
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• RefWorks (Direct export to RefWorks)

Short-term load forecasting (STLF) plays a key role in the formulation of economic, reliable, and secure operating strategies (planning, scheduling, maintenance, and control processes, among others) for a power system and will be significant in the future. However, there is still much to do in these research areas. The deployment of enabling technologies (e.g., smart meters) has made high-granularity data available for many customer segments and to approach many issues, for instance, to make forecasting tasks feasible at several demand aggregation levels. The first challenge is the improvement of STLF models and their performance at new aggregation levels. Moreover, the mix of renewables in the power system, and the necessity to include more flexibility through demand response initiatives have introduced greater uncertainties, which means new challenges for STLF in a more dynamic power system in the 2030–50 horizon. Many techniques have been proposed and applied for STLF, including traditional statistical models and AI techniques. Besides, distribution planning needs, as well as grid modernization, have initiated the development of hierarchical load forecasting. Analogously, the need to face new sources of uncertainty in the power system is giving more importance to probabilistic load forecasting. This Special Issue deals with both fundamental research and practical application research on STLF methodologies to face the challenges of a more distributed and customer-centered power system.

History of engineering & technology --- short-term load forecasting --- demand-side management --- pattern similarity --- hierarchical short-term load forecasting --- feature selection --- weather station selection --- load forecasting --- special days --- regressive models --- electric load forecasting --- data preprocessing technique --- multiobjective optimization algorithm --- combined model --- Nordic electricity market --- electricity demand --- component estimation method --- univariate and multivariate time series analysis --- modeling and forecasting --- deep learning --- wavenet --- long short-term memory --- demand response --- hybrid energy system --- data augmentation --- convolution neural network --- residential load forecasting --- forecasting --- time series --- cubic splines --- real-time electricity load --- seasonal patterns --- Load forecasting --- VSTLF --- bus load forecasting --- DBN --- PSR --- distributed energy resources --- prosumers --- building electric energy consumption forecasting --- cold-start problem --- transfer learning --- multivariate random forests --- random forest --- electricity consumption --- lasso --- Tikhonov regularization --- load metering --- preliminary load --- short term load forecasting --- performance criteria --- power systems --- cost analysis --- day ahead --- feature extraction --- deep residual neural network --- multiple sources --- electricity --- short-term load forecasting --- demand-side management --- pattern similarity --- hierarchical short-term load forecasting --- feature selection --- weather station selection --- load forecasting --- special days --- regressive models --- electric load forecasting --- data preprocessing technique --- multiobjective optimization algorithm --- combined model --- Nordic electricity market --- electricity demand --- component estimation method --- univariate and multivariate time series analysis --- modeling and forecasting --- deep learning --- wavenet --- long short-term memory --- demand response --- hybrid energy system --- data augmentation --- convolution neural network --- residential load forecasting --- forecasting --- time series --- cubic splines --- real-time electricity load --- seasonal patterns --- Load forecasting --- VSTLF --- bus load forecasting --- DBN --- PSR --- distributed energy resources --- prosumers --- building electric energy consumption forecasting --- cold-start problem --- transfer learning --- multivariate random forests --- random forest --- electricity consumption --- lasso --- Tikhonov regularization --- load metering --- preliminary load --- short term load forecasting --- performance criteria --- power systems --- cost analysis --- day ahead --- feature extraction --- deep residual neural network --- multiple sources --- electricity

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It is very well known that differential equations are related with the rise of physical science in the last several decades and they are used successfully for models of real-world problems in a variety of fields from several disciplines. Additionally, difference equations represent the discrete analogues of differential equations. These types of equations started to be used intensively during the last several years for their multiple applications, particularly in complex chaotic behavior. A certain class of differential and related difference equations is represented by their respective fractional forms, which have been utilized to better describe non-local phenomena appearing in all branches of science and engineering. The purpose of this book is to present some common results given by mathematicians together with physicists, engineers, as well as other scientists, for whom differential and difference equations are valuable research tools. The reported results can be used by researchers and academics working in both pure and applied differential equations.

Research & information: general --- Mathematics & science --- dynamic equations --- time scales --- classification --- existence --- necessary and sufficient conditions --- fractional calculus --- triangular fuzzy number --- double-parametric form --- FRDTM --- fractional dynamical model of marriage --- approximate controllability --- degenerate evolution equation --- fractional Caputo derivative --- sectorial operator --- fractional symmetric Hahn integral --- fractional symmetric Hahn difference operator --- Arrhenius activation energy --- rotating disk --- Darcy–Forchheimer flow --- binary chemical reaction --- nanoparticles --- numerical solution --- fractional differential equations --- two-dimensional wavelets --- finite differences --- fractional diffusion-wave equation --- fractional derivative --- ill-posed problem --- Tikhonov regularization method --- non-linear differential equation --- cubic B-spline --- central finite difference approximations --- absolute errors --- second order differential equations --- mild solution --- non-instantaneous impulses --- Kuratowski measure of noncompactness --- Darbo fixed point --- multi-stage method --- multi-step method --- Runge–Kutta method --- backward difference formula --- stiff system --- numerical solutions --- Riemann-Liouville fractional integral --- Caputo fractional derivative --- fractional Taylor vector --- kerosene oil-based fluid --- stagnation point --- carbon nanotubes --- variable thicker surface --- thermal radiation --- differential equations --- symmetric identities --- degenerate Hermite polynomials --- complex zeros --- oscillation --- third order --- mixed neutral differential equations --- powers of stochastic Gompertz diffusion models --- powers of stochastic lognormal diffusion models --- estimation in diffusion process --- stationary distribution and ergodicity --- trend function --- application to simulated data --- n-th order linear differential equation --- two-point boundary value problem --- Green function --- linear differential equation --- exponential stability --- linear output feedback --- stabilization --- uncertain system --- nonlocal effects --- linear control system --- Hilbert space --- state feedback control --- exact controllability --- upper Bohl exponent

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• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

It is very well known that differential equations are related with the rise of physical science in the last several decades and they are used successfully for models of real-world problems in a variety of fields from several disciplines. Additionally, difference equations represent the discrete analogues of differential equations. These types of equations started to be used intensively during the last several years for their multiple applications, particularly in complex chaotic behavior. A certain class of differential and related difference equations is represented by their respective fractional forms, which have been utilized to better describe non-local phenomena appearing in all branches of science and engineering. The purpose of this book is to present some common results given by mathematicians together with physicists, engineers, as well as other scientists, for whom differential and difference equations are valuable research tools. The reported results can be used by researchers and academics working in both pure and applied differential equations.

Research & information: general --- Mathematics & science --- dynamic equations --- time scales --- classification --- existence --- necessary and sufficient conditions --- fractional calculus --- triangular fuzzy number --- double-parametric form --- FRDTM --- fractional dynamical model of marriage --- approximate controllability --- degenerate evolution equation --- fractional Caputo derivative --- sectorial operator --- fractional symmetric Hahn integral --- fractional symmetric Hahn difference operator --- Arrhenius activation energy --- rotating disk --- Darcy–Forchheimer flow --- binary chemical reaction --- nanoparticles --- numerical solution --- fractional differential equations --- two-dimensional wavelets --- finite differences --- fractional diffusion-wave equation --- fractional derivative --- ill-posed problem --- Tikhonov regularization method --- non-linear differential equation --- cubic B-spline --- central finite difference approximations --- absolute errors --- second order differential equations --- mild solution --- non-instantaneous impulses --- Kuratowski measure of noncompactness --- Darbo fixed point --- multi-stage method --- multi-step method --- Runge–Kutta method --- backward difference formula --- stiff system --- numerical solutions --- Riemann-Liouville fractional integral --- Caputo fractional derivative --- fractional Taylor vector --- kerosene oil-based fluid --- stagnation point --- carbon nanotubes --- variable thicker surface --- thermal radiation --- differential equations --- symmetric identities --- degenerate Hermite polynomials --- complex zeros --- oscillation --- third order --- mixed neutral differential equations --- powers of stochastic Gompertz diffusion models --- powers of stochastic lognormal diffusion models --- estimation in diffusion process --- stationary distribution and ergodicity --- trend function --- application to simulated data --- n-th order linear differential equation --- two-point boundary value problem --- Green function --- linear differential equation --- exponential stability --- linear output feedback --- stabilization --- uncertain system --- nonlocal effects --- linear control system --- Hilbert space --- state feedback control --- exact controllability --- upper Bohl exponent --- dynamic equations --- time scales --- classification --- existence --- necessary and sufficient conditions --- fractional calculus --- triangular fuzzy number --- double-parametric form --- FRDTM --- fractional dynamical model of marriage --- approximate controllability --- degenerate evolution equation --- fractional Caputo derivative --- sectorial operator --- fractional symmetric Hahn integral --- fractional symmetric Hahn difference operator --- Arrhenius activation energy --- rotating disk --- Darcy–Forchheimer flow --- binary chemical reaction --- nanoparticles --- numerical solution --- fractional differential equations --- two-dimensional wavelets --- finite differences --- fractional diffusion-wave equation --- fractional derivative --- ill-posed problem --- Tikhonov regularization method --- non-linear differential equation --- cubic B-spline --- central finite difference approximations --- absolute errors --- second order differential equations --- mild solution --- non-instantaneous impulses --- Kuratowski measure of noncompactness --- Darbo fixed point --- multi-stage method --- multi-step method --- Runge–Kutta method --- backward difference formula --- stiff system --- numerical solutions --- Riemann-Liouville fractional integral --- Caputo fractional derivative --- fractional Taylor vector --- kerosene oil-based fluid --- stagnation point --- carbon nanotubes --- variable thicker surface --- thermal radiation --- differential equations --- symmetric identities --- degenerate Hermite polynomials --- complex zeros --- oscillation --- third order --- mixed neutral differential equations --- powers of stochastic Gompertz diffusion models --- powers of stochastic lognormal diffusion models --- estimation in diffusion process --- stationary distribution and ergodicity --- trend function --- application to simulated data --- n-th order linear differential equation --- two-point boundary value problem --- Green function --- linear differential equation --- exponential stability --- linear output feedback --- stabilization --- uncertain system --- nonlocal effects --- linear control system --- Hilbert space --- state feedback control --- exact controllability --- upper Bohl exponent