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This paper introduces a fast and numerically stable algorithm for the solution of fourth-order linear boundary value problems on an interval. This type of equation arises in a variety of settings in physics and signal processing. Our method…
The purpose of the present work is to study the existence of solutions to initial value problems for nonlinear first order differential systems with nonlinear nonlocal boundary conditions of functional type. The existence results are…
When the nonconvex problem is complicated by stochasticity, the sample complexity of stochastic first-order methods may depend linearly on the problem dimension, which is undesirable for large-scale problems. To alleviate this linear…
The nonlinear space-fractional problems often allow multiple stationary solutions, which can be much more complicated than the corresponding integer-order problems. In this paper, we systematically compute the solution landscapes of…
We prove that a semidiscrete $(2r+1)$-point scheme for quasilinear first order PDE cannot attain an order higher than $2r$. Moreover, if the forward Euler fully discrete scheme obtained from the linearization about any constant state of the…
A new (algebraic) approximation scheme to find {\sl global} solutions of two point boundary value problems of ordinary differential equations (ODE's) is presented. The method is applicable for both linear and nonlinear (coupled) ODE's whose…
This chapter presents some numerical methods to solve problems in the fractional calculus of variations and fractional optimal control. Although there are plenty of methods available in the literature, we concentrate mainly on approximating…
This work is concerned with the development of quasi-Trefftz methods for first-order differential systems. It focuses on discrete quasi-Trefftz spaces, starting from their definition and including the construction of corresponding bases…
We present a novel approach for high-order accurate numerical differentiation on unstructured meshes of quadrilateral elements. To differentiate a given function, an auxiliary function with greater smoothness properties is defined which…
We derive several numerical methods for designing optimized first-order algorithms in unconstrained convex optimization settings. Our methods are based on the Performance Estimation Problem (PEP) framework, which casts the worst-case…
We introduce a predictor-corrector discretisation scheme for the numerical integration of a class of stochastic differential equations and prove that it converges with weak order 1.0. The key feature of the new scheme is that it builds up…
We prove a number of \textit{a priori} estimates for weak solutions of elliptic equations or systems with vertically independent coefficients in the upper-half space. These estimates are designed towards applications to boundary value…
This paper presents a novel approach for numerical solution of a class of fourth order time fractional partial differential equations (PDE's). The finite difference formulation has been used for temporal discretization, whereas, the space…
A class of second order approximations, called the weighted and shifted Gr\"{u}nwald difference operators, are proposed for Riemann-Liouville fractional derivatives, with their effective applications to numerically solving space fractional…
System of semilinear ordinary differential equation and fractional differential equation of distributed order is investigated and solved in a mild and classical sense. Such a system arises as a distributed derivative model of…
We introduce basic aspects of new operator method, which is very suitable for practical solving differential equations of various types. The main advantage of the method is revealed in opportunity to find compact exact operator solutions of…
This work develops a framework to discover relations between the components of the solution to a given initial-value problem for a first-order system of ordinary differential equations. This is done by using sparse identification techniques…
This paper is devoted to studying the first-order variational analysis of non-convex and non-differentiable functions that may not be subdifferentially regular. To achieve this goal, we entirely rely on two concepts of directional…
A sequential quadratic programming method is designed for solving general smooth nonlinear stochastic optimization problems subject to expectation equality constraints. We consider the setting where the objective and constraint function…
Sticky diffusion models a Markovian particle experiencing reflection and temporary adhesion phenomena at the boundary. Numerous numerical schemes exist for approximating stopped or reflected stochastic differential equations (SDEs), but…