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### Extending the deterministic Riemann–Liouville and Caputo operators to the random framework: A mean square approach with applications to solve random fractional differential equations

C. Burgos, J.-C. Cortés, L. Villafuerte, R.-J. Villanueva
2017 Chaos, Solitons & Fractals
Extending the deterministic Riemann-Liouville and Caputo operators to the random framework: A mean square approach with applications to solve random fractional differential equations.  ...  Keywords: Random mean square Riemann-Liouville integral, random mean square Caputo derivative, random fractional linear differential equation, random Fröbenius method.  ...  Solving the random linear fractional differential equation by the mean square general- 262 ized Fröbenius method 263 This section is devoted to construct a solution SP to the random fractional linear differential  ...

### Analytic-Numerical Solution of Random Parabolic Models: A Mean Square Fourier Transform Approach

Maria-Consuelo Casaban, Juan-Carlos Cortes, Lucas Jodar
2018 Mathematical Modelling and Analysis
This paper deals with the construction of mean square analytic-numerical solution of parabolic partial differential problems where both initial condition and coefficients are stochastic processes.  ...  By using a random Fourier transform, an inf- nite integral form of the solution stochastic process is firstly obtained.  ...  partial differential initial value problems where the uncertainty is treated in the mean square sense.  ...

### Linear Partial Differential Equations with Random Forcing

Frederic Y. M. Wan
1972 Studies in applied mathematics (Cambridge)
In any event, the fact that equations (1.9) and (1.10) form a closed system is only a special feature of (1.7) and not true in general for (1.1).  ...  .\$) alone appears in (1.9) and (1.10) so that these become two equations for this sum and (u&. But there does not seem to be any way for us to determine (u:) and (u;) separately.  ...  Acknowledgements The author is grateful to his colleague, Professor Steven Orszag, who read and commented on the manuscript.  ...

### Numerical Approximation of Partial Differential Equations by a Variable Projection Method with Artificial Neural Networks [article]

Suchuan Dong, Jielin Yang
2022 arXiv   pre-print
For linear PDEs, enforcing the boundary/initial value problem on the collocation points leads to a separable nonlinear least squares problem about the network coefficients.  ...  For nonlinear PDEs, enforcing the boundary/initial value problem on the collocation points leads to a nonlinear least squares problem that is not separable, which precludes the VarPro strategy for such  ...  Given a nonlinear boundary/initial value problem, we first linearize the problem for the Newton iteration, with a particular linearized form.  ...

### A Deep Neural Network Surrogate for High-Dimensional Random Partial Differential Equations [article]

2018 arXiv   pre-print
Specifically, the random PDE is approximated by a feed-forward fully-connected deep residual network, with either strong or weak enforcement of initial and boundary constraints.  ...  Developing efficient numerical algorithms for the solution of high dimensional random Partial Differential Equations (PDEs) has been a challenging task due to the well-known curse of dimensionality.  ...  and boundary conditions, e.g. by using Lagrange multipliers, or by developing algorithms for learning the functions C and G in Equation 11, and (4) investigating optimal sampling strategies in order to  ...

### Differential equations with nonlinear boundary conditions

G.L Cohen
1977 Journal of Mathematical Analysis and Applications
Notice that in the more standard two-point boundary value problem with tz linear boundary conditions, the algebraic system above becomes a system of 2n linear equations in 2n unknowns.  ...  Under their convention, Edwards and Moyal solved (21) in a Banach space setting, subject to linear two-point boundary conditions and with certain conditions attached to x.  ...

### Fast integral equation methods for the heat equation and the modified Helmholtz equation in two dimensions [article]

Mary-Catherine Kropinski, Bryan Quaife
2010 arXiv   pre-print
We present an efficient integral equation approach to solve the heat equation, u_t () - Δ u() = F(,t), in a two-dimensional, multiply connected domain, and with Dirichlet boundary conditions.  ...  For a total of N points in the discretization of the boundary and the domain, the total computational cost per time step is O(N) or O(N N).  ...  Acknowledgements: We wish to thank the organizers of the conference in Advances in Boundary Integral Equations and Related Topics for their invitation and the opportunity to participate in this special  ...

### Understanding differential equations through diffusion point of view [article]

Dohy Hong
2012 arXiv   pre-print
In this paper, we propose a new adaptation of the D-iteration algorithm to numerically solve the differential equations.  ...  This problem can be reinterpreted in 2D or 3D (or higher dimensions) as a limit of a diffusion process where the boundary or initial conditions are replaced by fluid catalysts.  ...  From the linear algebra point of view, this means that we have a matrix associated to the equation (6) with initial condition g(0) = 1 and that its spectral radius is strictly less than 1 thanks to the  ...

### Radial function collocation solution of partial differential equations in irregular domains

V. Pereyra, G. Scherer, P. Gonzalez Casanova
2007 International Journal of Computing Science and Mathematics
We use a regularised least squares approach to solve the potentially ill-conditioned problems that may arise.  ...  We consider a collocation method using radial functions for the solution of partial differential equations in irregular domains.  ...  values and using VARPRO to solve the resulting separable non-linear least squares problem.  ...

### Solving the Nonlinear Two-Dimension Wave Equation Using Dual Reciprocity Boundary Element Method

Kumars Mahmoodi, Hassan Ghassemi, Alireza Heydarian
2017 International Journal of Partial Differential Equations and Applications
This paper presents to solve the nonlinear 2-D wave equation defined over a rectangular spatial domain with appropriate initial and boundary conditions.  ...  Two-dimension wave equation is a time-domain problem, with three independent variables , , .  ...  Consider the th order linear PDE in just two independent spatial variables , and time , with coefficients independentof time, defined for > 0 over the finite region ∈ 2 , with sufficient initial and boundary  ...

### ODEN: A Framework to Solve Ordinary Differential Equations using Artificial Neural Networks [article]

Liam L.H. Lau, Denis Werth
2020 arXiv   pre-print
We explore in detail a method to solve ordinary differential equations using feedforward neural networks.  ...  However, we observe an optimal architecture that matches the complexity of the differential equation. A user-friendly and adaptable open-source code (ODEN) is provided on GitHub.  ...  The loss function contains two terms: the first one forces the differential equation (3) to hold, and the second one encodes the boundary/initial conditions 3 .  ...

### Uncertainty Quantification in Neural Differential Equations [article]

Olga Graf, Pablo Flores, Pavlos Protopapas, Karim Pichara
2021 arXiv   pre-print
Among applications that can benefit from effective handling of uncertainty are the deep learning based differential equation (DE) solvers.  ...  Uncertainty quantification (UQ) helps to make trustworthy predictions based on collected observations and uncertain domain knowledge.  ...  Given the computational expense of sampling during Bayesian NN training, the latter two methods could be preferable in the case of complex DEs on a multidimentional domain.  ...

### Explicit Estimation of Derivatives from Data and Differential Equations by Gaussian Process Regression [article]

Hongqiao Wang, Xiang Zhou
2020 arXiv   pre-print
In this work, we employ the Bayesian inference framework to solve the problem of estimating the solution and particularly, its derivatives, which satisfy a known differential equation, from the given noisy  ...  By regarding the linear differential equation as a linear constraint, a Gaussian process regression with constraint method (GPRC) is developed to improve the accuracy of prediction of derivatives.  ...  We define a normal distribution p(u|x, IC/BC) which contains the IC/BC information, whose mean is the state value at the point x 0 (the nearest initial or boundary point to x) and whose variance increases  ...

### A new approach to solving multi-order fractional equations using BEM and Chebyshev matrix [article]

Moein Khalighi, Mohammad Amirian Matlob, Alaeddin Malek
2020 arXiv   pre-print
In this paper, the boundary element method is combined with Chebyshev operational matrix technique to solve two-dimensional multi-order time-fractional partial differential equations; nonlinear and linear  ...  Boundary element method is used to convert the main problem into a system of a multi-order fractional ordinary differential equation.  ...  ACKNOWLEDGEMENTS The authors are very grateful to the referees for carefully reading the paper and for their comments and suggestions which have improved the paper.  ...

### Complex Langevin equations and Schwinger–Dyson equations

Gerald Guralnik, Cengiz Pehlevan
2009 Nuclear Physics B
Specific examples in zero dimensions and on a lattice are given. Relevance to the study of quantum field theory phase space is discussed.  ...  Stationary distributions of complex Langevin equations are shown to be the complexified path integral solutions of the Schwinger-Dyson equations of the associated quantum field theory.  ...  Ferrante and D. Obeid for useful discussions and conversations. We thank the anonymous reviewer for urging us to clarify some points and bringing references [33] and [34] to our attention. G.  ...
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