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On the properties of the exceptional set for the randomized Euler and Runge-Kutta schemes
We show that the probability of the exceptional set decays exponentially for a broad class of randomized algorithms approximating solutions of ODEs,...
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An accurate and efficient space-time Galerkin spectral method for the subdiffusion equation
In this paper, we design and analyze a space-time spectral method for the subdiffusion equation. Here, we are facing two difficulties. The first is...
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A new approach to proper orthogonal decomposition with difference quotients
In a recent work (Koc et al., SIAM J. Numer. Anal. 59 (4), 2163–2196,
2021 ), the authors showed that including difference quotients (DQs) is necessary... -
The difference schemes for solving singularly perturbed three-point boundary value problem
In this paper, we propose and analyze numerical treatment for a singularly perturbed convection–diffusion boundary value problem with nonlocal...
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An hp-version of the C0-continuous Petrov-Galerkin time step** method for nonlinear second-order initial value problems
We present an hp -version of the C 0 -continuous Petrov-Galerkin time step** method for nonlinear second-order initial value problems. We derive a...
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Fractional-order Bessel functions with various applications
We introduce fractional-order Bessel functions (FBFs) to obtain an approximate solution for various kinds of differential equations. Our main aim is...
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Rigorous and effective a-posteriori error bounds for nonlinear problems—application to RB methods
Quantifying the error that is induced by numerical approximation techniques is an important task in many fields of applied mathematics. Two...
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The Virtual Element Method for Eigenvalue Problems with Potential Terms on Polytopic Meshes
We extend the conforming virtual element method (VEM) to the numerical resolution of eigenvalue problems with potential terms on a polytopic mesh. An...
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Adaptive mesh selection asymptotically guarantees a prescribed local error for systems of initial value problems
We study potential advantages of adaptive mesh point selection for the solution of systems of initial value problems. For an optimal order...
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Uniform and high-order discretization schemes for Sturm–Liouville problems via Fer streamers
The current paper concerns the uniform and high-order discretization of the novel approach to the computation of Sturm–Liouville problems via Fer...
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An h-p version of the continuous Petrov-Galerkin method for Volterra delay-integro-differential equations
We consider an h - p version of the continuous Petrov-Galerkin time step** method for Volterra integro-differential equations with proportional...
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A fully diagonalized spectral method using generalized Laguerre functions on the half line
A fully diagonalized spectral method using generalized Laguerre functions is proposed and analyzed for solving elliptic equations on the half line....
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Generalized polynomial chaos for nonlinear random pantograph equations
This paper is concerned with the application of generalized polynomial chaos (gPC) method to nonlinear random pantograph equations. An error...
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Optimal L ∞ estimates for Galerkin methods for nonlinear singular two-point boundary value problems
In this paper, we apply the symmetric Galerkin methods to the numerical solutions of a kind of singular linear two-point boundary value problems. We...
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On the constructive investigation of a class of linear boundary value problems for n th order differential equations with deviating arguments
A constructive technique for the study of boundary value problems for n th order differential equations with deviating arguments is described. A...
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The new modified Ishikawa iteration method for the approximate solution of different types of differential equations
In this article, the new Ishikawa iteration method is presented to find the approximate solution of an ordinary differential equation with an initial...
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On the optimal harvesting of size-structured population dynamics
This work is concerned with a kind of optimal control problem for a size-structured biological population model. Well-posedness of the state system...
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Convergence rates and explicit error bounds of Hill’s method for spectra of self-adjoint differential operators
We present the convergence rates and the explicit error bounds of Hill’s method, which is a numerical method for computing the spectra of ordinary...
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Stochastic Galerkin techniques for random ordinary differential equations
Over the last decade the stochastic Galerkin method has become an established method to solve differential equations involving uncertain parameters....
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Three-points interfacial quadrature for geometrical source terms on nonuniform grids
This paper deals with numerical (finite volume) approximations, on nonuniform meshes, for ordinary differential equations with parameter-dependent...