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Various traditional numerical methods for solving initial value problems of differential equations often produce local solutions near the initial value point, despite the problems having larger interval solutions. Even current popular…
We study the following ultraparabolic equation \[ \frac{\partial}{\partial t}u\left(t,s\right)+\frac{\partial}{\partial…
Mathematical analysis with numerical application of the newly formulated fractional version of the Adams-Bashforth method using the Atangana-Baleanu derivative which has nonlocal and nonsingular properties is considered in this paper. We…
We introduce a framework for quasi-Newton forward--backward splitting algorithms (proximal quasi-Newton methods) with a metric induced by diagonal $\pm$ rank-$r$ symmetric positive definite matrices. This special type of metric allows for a…
A new approach for integration of the initial value problem for ordinary differential equations is suggested. The algorithm is based on approximation of the solution by a system of functions that contains orthogonal exponential polynomials.
We study a probabilistic numerical method for the solution of both boundary and initial value problems that returns a joint Gaussian process posterior over the solution. Such methods have concrete value in the statistics on Riemannian…
We study some functional inequalities satisfied by the distribution of the solution of a stochastic differential equation driven by fractional Brownian motions. Such functional inequalities are obtained through new integration by parts…
In the framework of fractional stochastic calculus, we study the existence and the uniqueness of the solution for a backward stochastic differential equation, formally written as: [{[c]{l}% -dY(t)= f(t,\eta(t),Y(t),Z(t))dt-Z(t)\delta…
This article examines a new approach to solving ordinary differential equations based on Fractional-Calculus theory. Poisson and Sturm-Liouville-type problems are studied, together with different boundary conditions. Each case is analyzed…
We consider the problem of approximating the solution of variational problems subject to the constraint that the admissible functions must be convex. This problem is at the interface between convex analysis, convex optimization, variational…
In engineering design, one often wishes to calculate the probability that the performance of a system is satisfactory under uncertainty. State of the art algorithms exist to solve this problem using active learning with Gaussian process…
This paper presents a novel method that allows to generalise the use of the Adam-Bashforth to Partial Differential Equations with local and non local operator. The Method derives a two step Adam-Bashforth numerical scheme in Laplace space…
In this paper, we determine the complex-valued solutions of the functional equation $$ f(x\sigma(y))+f(\tau(y)x)=2f(x)f(y)$$ for all $x,y \in M$, where $M$ is a monoid, $\sigma$: $M\longrightarrow M$ is an involutive automorphism and…
An equation containing a fractional power of an elliptic operator of second order is studied for Dirichlet boundary conditions. Finite difference approximations in space are employed. The proposed numerical algorithm is based on solving an…
This paper presents a new algorithm, termed \emph{truncated amplitude flow} (TAF), to recover an unknown vector $\bm{x}$ from a system of quadratic equations of the form $y_i=|\langle\bm{a}_i,\bm{x}\rangle|^2$, where $\bm{a}_i$'s are given…
In this paper we present a numerical solution of a one-phase 1D fractional Stefan problem with Caputo derivative with respect to time variable. In the proposed approach, we use a front fixing method and the algorithm of numerical…
In this paper we are interested in the numerical solution of stochastic differential equations with non negative solutions. Our goal is to construct explicit numerical schemes that preserve positivity, even for super linear stochastic…
The following document presents some novel numerical methods valid for one and several variables, which using the fractional derivative, allow to find solutions for some non-linear systems in the complex space using real initial conditions.…
In this note, a numerical method based on finite differences to solve a class of nonlinear advection-diffusion fractional differential equation is proposed. The fractional operator considered here is the fractional Riemann-Liouville…
In this article we introduce a simple straightforward and powerful method involving symbolic manipulation, Picard iteration, and auxiliary variables for approximating solutions of partial differential boundary value problems. The method is…