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Runge-Kutta formulas. --- Runge-Kutta methods --- Differential equations --- Numerical solutions --- Fórmules de Runge-Kutta --- Mètodes de Runge-Kutta

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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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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.

Information technology industries --- 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 --- 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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"This book describes numerical methods for partial differential equations (PDEs) coupling advection, diffusion and reaction terms, encompassing methods for hyperbolic, parabolic and stiff and nonstiff ordinary differential equations (ODEs). The emphasis lies on time-dependent transport-chemistry problems, describing e.g. the evolution of concentrations in environmental and biological applications. Along with the common topics of stability and convergence, much attention is paid on how to prevent spurious, negative concentrations and oscillations, both in space and time. Many of the theoretical aspects are illustrated by numerical experiments on models from biology, chemistry and physics. A unified approach is followed by emphasizing the method of lines or semi-discretization. In this regard this book differs substantially from more specialized textbooks which deal exclusively with either PDEs or ODEs. This book treats integration methods suitable for both classes of problems and thus is of interest to PDE researchers unfamiliar with advanced numerical ODE methods, as well as to ODE researchers unaware of the vast amount of interesting results on numerical PDEs". -- Cover.

Differential equations --- Differential equations, Partial --- Stiff computation (Differential equations) --- Runge-Kutta formulas --- Numerical solutions --- -Differential equations, Partial --- -Stiff computation (Differential equations) --- Runge-Kutta methods --- Computation, Stiff (Differential equations) --- Equations, Stiff (Differential equations) --- Stiff equations (Differential equations) --- Stiff systems (Differential equations) --- Systems, Stiff (Differential equations) --- Runge-Kutta formulas. --- Stiff computation (Differential equations). --- 519.63 --- 519.63 Numerical methods for solution of partial differential equations --- Numerical methods for solution of partial differential equations --- 517.91 Differential equations --- Partial differential equations --- Numerical solutions of differential equations --- 517.91 --- Numerical solutions. --- Applied mathematics. --- Engineering mathematics. --- Partial differential equations. --- Differential equations. --- Numerical analysis. --- Mathematical and Computational Engineering. --- Partial Differential Equations. --- Ordinary Differential Equations. --- Numerical Analysis. --- Mathematical analysis --- Engineering --- Engineering analysis --- Mathematics --- Numerical solutions&delete& --- Numerical analysis --- Differential equations - Numerical solutions --- Differential equations, Partial - Numerical solutions

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Many industries, such as transportation and manufacturing, use control systems to insure that parameters such as temperature or altitude behave in a desirable way over time. For example, pilots need assurance that the plane they are flying will maintain a particular heading. An integral part of control systems is a mechanism for failure detection to insure safety and reliability. This book offers an alternative failure detection approach that addresses two of the fundamental problems in the safe and efficient operation of modern control systems: failure detection--deciding when a failure has occurred--and model identification--deciding which kind of failure has occurred. Much of the work in both categories has been based on statistical methods and under the assumption that a given system was monitored passively. Campbell and Nikoukhah's book proposes an "active" multimodel approach. It calls for applying an auxiliary signal that will affect the output so that it can be used to easily determine if there has been a failure and what type of failure it is. This auxiliary signal must be kept small, and often brief in duration, in order not to interfere with system performance and to ensure timely detection of the failure. The approach is robust and uses tools from robust control theory. Unlike some approaches, it is applicable to complex systems. The authors present the theory in a rigorous and intuitive manner and provide practical algorithms for implementation of the procedures.

System failures (Engineering) --- Fault location (Engineering) --- Signal processing. --- Processing, Signal --- Information measurement --- Signal theory (Telecommunication) --- Location of system faults --- System fault location (Engineering) --- Dynamic testing --- Failure of engineering systems --- Reliability (Engineering) --- Systems engineering --- A priori estimate. --- AIXI. --- Abuse of notation. --- Accuracy and precision. --- Additive white Gaussian noise. --- Algorithm. --- Approximation. --- Asymptotic analysis. --- Bisection method. --- Boundary value problem. --- Calculation. --- Catastrophic failure. --- Combination. --- Computation. --- Condition number. --- Continuous function. --- Control theory. --- Control variable. --- Decision theory. --- Derivative. --- Detection. --- Deterministic system. --- Diagram (category theory). --- Differential equation. --- Discrete time and continuous time. --- Discretization. --- Dynamic programming. --- Engineering design process. --- Engineering. --- Equation. --- Error message. --- Estimation theory. --- Estimation. --- Finite difference. --- Gain scheduling. --- Inequality (mathematics). --- Initial condition. --- Integrator. --- Invertible matrix. --- Laplace transform. --- Least squares. --- Likelihood function. --- Likelihood-ratio test. --- Limit point. --- Linear programming. --- Linearization. --- Mathematical optimization. --- Mathematical problem. --- Maxima and minima. --- Measurement. --- Method of lines. --- Monotonic function. --- Noise power. --- Nonlinear control. --- Nonlinear programming. --- Norm (mathematics). --- Numerical analysis. --- Numerical control. --- Numerical integration. --- Observational error. --- Open problem. --- Optimal control. --- Optimization problem. --- Parameter. --- Partial differential equation. --- Piecewise. --- Pointwise. --- Prediction. --- Probability. --- Random variable. --- Realizability. --- Remedial action. --- Requirement. --- Rewriting. --- Riccati equation. --- Runge–Kutta methods. --- Sampled data systems. --- Sampling (signal processing). --- Scientific notation. --- Scilab. --- Shift operator. --- Signal (electrical engineering). --- Sine wave. --- Solver. --- Special case. --- Stochastic Modeling. --- Stochastic calculus. --- Stochastic interpretation. --- Stochastic process. --- Stochastic. --- Theorem. --- Time complexity. --- Time-invariant system. --- Trade-off. --- Transfer function. --- Transient response. --- Uncertainty. --- Utilization. --- Variable (mathematics). --- Variance.