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Analysis of a Narrow-Stencil Finite Difference Method for Approximating Viscosity Solutions of Fully Nonlinear Second Order Parabolic PDEs
This paper approximates viscosity solutions of fully nonlinear second order parabolic PDEs by a narrow-stencil finite difference...
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Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions
Nonlinear partial differential equations (PDEs) are used to model dynamical processes in a large number of scientific fields, ranging from finance to...
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State-dependent Riccati equation feedback stabilization for nonlinear PDEs
The synthesis of suboptimal feedback laws for controlling nonlinear dynamics arising from semi-discretized PDEs is studied. An approach based on the...
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Fully Nonlinear Equations
In this chapter, we develop the theory of fully nonlinear nonlocal elliptic equations. We begin with the definition of viscosity solutions and their...
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Survey on Path-Dependent PDEs
In this paper, the authors provide a brief introduction of the path-dependent partial di.erential equations (PDEs for short) in the space of...
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Neural networks-based backward scheme for fully nonlinear PDEs
We propose a numerical method for solving high dimensional fully nonlinear partial differential equations (PDEs). Our algorithm estimates...
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Learning-Informed Parameter Identification in Nonlinear Time-Dependent PDEs
We introduce and analyze a method of learning-informed parameter identification for partial differential equations (PDEs) in an all-at-once...
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A fully nonlinear Feynman–Kac formula with derivatives of arbitrary orders
We present an algorithm for the numerical solution of nonlinear parabolic partial differential equations. This algorithm extends the classical...
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New High-Order Numerical Methods for Hyperbolic Systems of Nonlinear PDEs with Uncertainties
In this paper, we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations (PDEs) with...
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Error estimates for POD-DL-ROMs: a deep learning framework for reduced order modeling of nonlinear parametrized PDEs enhanced by proper orthogonal decomposition
POD-DL-ROMs have been recently proposed as an extremely versatile strategy to build accurate and reliable reduced order models (ROMs) for nonlinear...
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Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions
We study fully nonlinear second-order (forward) stochastic PDEs. They can also be viewed as forward path-dependent PDEs and will be treated as rough...
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An energy formula for fully nonlinear degenerate parabolic equations in one spatial dimension
Energy (or Lyapunov) functions are used to prove stability of equilibria, or to indicate a gradient-like structure of a dynamical system. Matano...
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Solving Time-Dependent PDEs with the Ultraspherical Spectral Method
We apply the ultraspherical spectral method to solving time-dependent PDEs by proposing two approaches to discretization based on the method of lines...
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Sturm attractors for fully nonlinear parabolic equations
We explicitly construct global attractors of fully nonlinear parabolic equations in one spatial dimension. These attractors are decomposed as...
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A nonlinear compact method based on double reduction order scheme for the nonlocal fourth-order PDEs with Burgers’ type nonlinearity
In this article, a novel double reduction order technique and a newly constructed nonlinear compact difference operator are developed on graded...
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First Order PDEs
The first equation is constant coefficient, the second equation is linear, the third equation quasilinear and the last equation nonlinear.
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Continuation and Bifurcation in Nonlinear PDEs – Algorithms, Applications, and Experiments
Numerical continuation and bifurcation methods can be used to explore the set of steady and time–periodic solutions of parameter dependent nonlinear...
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Coordinate-Adaptive Integration of PDEs on Tensor Manifolds
We introduce a new tensor integration method for time-dependent partial differential equations (PDEs) that controls the tensor rank of the PDE...
