Random Boundary Value Research Articles

Sparse series solutions of random boundary and initial value problems

The study investigates the utilization of sparse representations through dictionary elements to address the classical Dirichlet and Cauchy types of stochastic boundary value problems (BVPs) and initial value problems (IVPs). A novel approach is introduced based on the recently developed stochastic pre-orthogonal adaptive Fourier decomposition (SPOAFD) technique. By employing SPOAFD, both analytic and numerical solutions for the stochastic BVPs and IVPs are formulated. Furthermore, the scope of the study is extended to include BVPs and IVPs associated with a specific class of fractional heat equations and fractional Poisson equations. In addition to establishing the theoretical framework, important computational aspects are thoroughly discussed to enable the implementation of practical algorithms. The proposed methodology is validated through numerical examples, demonstrating its effectiveness and computational efficiency.

On the Caputo fractional random boundary value problem

On the Caputo fractional random boundary value problem

The existence of positive solutions for φ-Hilfer fractional differential equation with random impulses and boundary value conditions

In this paper, we consider a class of impulsive fractional differential equations involving φ-Hilfer fractional derivatives. Since the φ-Hilfer fractional derivative depends on the kernel φ, for a suitable choice of the kernel φ, our results in this paper can be applied to most of the fractional derivatives. First, we give a correspondence in this article between our problem and a Volterra-type integral equation. Then, sufficient conditions are given to ensure the existence of positive solutions and by applying the cone stretch and cone compression immobilization theorem and the Green function to study the existence of positive solutions for the problem. Finally, an example is given to illustrate the viability of the main theories.

The existence of solutions for Sturm\u2013Liouville differential equation with random impulses and boundary value problems

In this article, we consider the existence of solutions to the Sturm–Liouville differential equation with random impulses and boundary value problems. We first study the Green function of the Sturm–Liouville differential equation with random impulses. Then, we get the equivalent integral equation of the random impulsive differential equation. Based on this integral equation, we use Dhage’s fixed point theorem to prove the existence of solutions to the equation, and the theorem is extended to the general second order nonlinear random impulsive differential equations. Then we use the upper and lower solution method to give a monotonic iterative sequence of the generalized random impulsive Sturm–Liouville differential equations and prove that it is convergent. Finally, we give two concrete examples to verify the correctness of the results.

Significance-Based Estimation-of-Distribution Algorithms

Estimation-of-distribution algorithms (EDAs) are randomized search heuristics that create a probabilistic model of the solution space, which is updated iteratively, based on the quality of the solutions sampled according to the model. As previous works show, this iteration-based perspective can lead to erratic updates of the model, in particular, to bit-frequencies approaching a random boundary value. In order to overcome this problem, we propose a new EDA based on the classic compact genetic algorithm (cGA) that takes into account a longer history of samples and updates its model only with respect to information which it classifies as statistically significant. We prove that this significance-based cGA (sig-cGA) optimizes the commonly regarded benchmark functions OneMax (OM), LeadingOnes, and BinVal all in quasilinear time, a result shown for no other EDA or evolutionary algorithm so far. For the recently proposed stable compact genetic algorithm—an EDA that tries to prevent erratic model updates by imposing a bias to the uniformly distributed model—we prove that it optimizes OM only in a time exponential in its hypothetical population size. Similarly, we show that the convex search algorithm cannot optimize OM in polynomial time.

Some notes on a second-order random boundary value problem

We consider a two-point boundary value problem of second-order random differential equation. Using a variant of the α-ψ-contractive type mapping theorem in metric spaces, we show the existence of at least one solution.

Numerical solution of the homogeneous Neumann boundary value problem on domains with a thin layer of random thickness

The present article is dedicated to the numerical solution of homogeneous Neumann boundary value problems on domains with a thin layer of different conductivity and of random thickness. By changing the boundary condition, the boundary value problem given on the random domain can be transformed into a boundary value problem on a fixed domain. The randomness is then contained in the coefficients of the new boundary condition. This thin coating can be expressed by a random Ventcell boundary condition and yields a second order accurate solution in the scale parameter ε of the layer's thickness. With the help of the Karhunen–Loève expansion, we transform this random boundary value problem into a deterministic, parametric one with a possibly high-dimensional parameter y. Based on the decay of the random fluctuations of the layer's thickness, we prove rates of decay of the derivatives of the random solution with respect to this parameter y which are robust in the scale parameter ε. Numerical results validate our theoretical findings.

Numerical Solution of the Poisson Equation on Domains with a Thin Layer of Random Thickness

The present article is dedicated to the numerical solution of the Poisson equation on domains with a thin layer of different conductivity and of random thickness. By changing the boundary condition, the boundary value problem given on a random domain is transformed into a boundary value problem on a fixed domain. The randomness is then contained in the coefficients of the new boundary condition. This thin coating can be expressed by a random Robin boundary condition which yields a third order accurate solution in the scale parameter $\varepsilon$ of the layer's thickness. With the help of the Karhunen--Loeve expansion, we transform this random boundary value problem into a deterministic parametric one with a possibly high-dimensional parameter ${\bf y}$. Based on the decay of the random fluctuations of the layer's thickness, we prove rates of decay of the derivatives of the random solution with respect to this parameter ${\bf y}$ which are robust in the scale parameter $\varepsilon$. Numerical results val...

Justification of Application of the Fourier Method for Parabolic Equations with Random Boundary Conditions from Orlicz Spaces

A new method of constructing solutions of boundary value problems for parabolic equations with random boundary value conditions is proposed. We assume that the boundary conditions are stochastic processes from the Orlicz space of random variables (including processes with zero mean values).

High-order methods as an alternative to using sparse tensor products for stochastic Galerkin FEM

Solutions of random elliptic boundary value problems admit efficient approximations by polynomials on the parameter domain. Each coefficient in such an expansion is a spatially dependent function, and can be approximated within a hierarchy of finite element spaces. If the finite elements are of sufficiently high order, using just a single spatial mesh is predicted to achieve the same convergence rate with respect to the total number of degrees of freedom as sparse tensor product constructions and other multilevel stochastic Galerkin approximations. Numerical computations for an elliptic two-point boundary value problem confirm this and indicate no loss of accuracy for a single-level method compared to using a sparse tensor product with the same total number of degrees of freedom.

Analytic-Numerical Solution of Random Boundary Value Heat Problems in a Semi-Infinite Bar

This paper deals with the analytic-numerical solution of random heat problems for the temperature distribution in a semi-infinite bar with different boundary value conditions. We apply a random Fourier sine and cosine transform mean square approach. Random operational mean square calculus is developed for the introduced transforms. Using previous results about random ordinary differential equations, a closed form solution stochastic process is firstly obtained. Then, expectation and variance are computed. Illustrative numerical examples are included.

On boundry value problems for random nonlinear differential equations

This paper deals with the nonlinear boundary value problem including random effects. An approximate probability distribution function is constructed by numerical integrated of a set of related deterministic problems. Convergence of the approximate distribution function in the actual distribution function is established. Key words: Random boundary value problem, distribution function, numerical integrated differential equation.

Error estimates for random boundary value problems with applications to a hanging cable problem

By employing Green's function approach, random effects in nonlinear boundary value problems are analyzed. In particular, by developing analytic techniques, estimates on the absolute mean and p th mean deviations of solutions of random boundary value problem from the smooth case are obtained. This methodology is applied to study the effects of randomness in hanging cable (telephone/power) problem. Furthermore, computational work is conducted to test and to compare the analytic results.

Moment Functions for Solutions of Random Boundary Value Problems

AbstractBoundary value problems for a class of ordinary differential operators with random coefficients are investigated. The random influences to the differential operators and the inhomogeneous terms are modelled by the class of weakly correlated processes. Using a perturbation method the random solutions can be represented as integral functionals of weakly correlated processes. It is possible to find approximations for the moment functions of the solutions by means of expansions in powers of the correlation length e of the input processes. Especially asymptotic expansions for second‐order moments of the solutions are presented. The results determined analytically are compared with Monte‐Carlo simulations.

A class of random variational inequalities and simple random unilateral boundary value problems - existence, discretization, finite element approximation

The paper deals with a class of random variational inequalities and simple random elliptic boundary value problems with unilateral conditions. Here randomness enters in the coefficient of the elliptic operator and in the right hand side of the p.d.e. In addition to existence and uniqueness results a theory of combined probabilistic deterministic discretization is developped that includes nonconforming approxima¬tion of unilateral constraints. Without any regularity assumptions on the solution, norm convergence of the full approximation process is established. The theory is applied to a Helmholtz like elliptic equation with Signorini boundary conditions as a simple model problem, where Galerkin discretization is realized by finite element approximation

The Computer-Oriented Calculus Course at Rensselaer Polytechnic Institute

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsWilliam E. BoyceWilliam E. Boyce has his B.A. from Rhodes College and his M.S. and Ph.D. (1955) from Carnegie-Mellon University, where his dissertation director was George Handelman. He now occupies the Edward P. Hamilton Chair of Science Education at Rensselaer Polytechnic Institute. Research interests include random and deterministic boundary value problems and their applications in continuum mechanics. With the late R. C. DiPrima he wrote Elementary Differential Equations and Boundary Value Problems, now in its fifth edition. He enjoys the Adirondack Mountains and following the fortunes of the St. Louis Cardinals. His wife, Elsa K. Boyce, is also a mathematician; they have three children and one grandchild.Joseph G. EckerJoe Ecker is a professor of mathematics at Rensselaer Polytechnic Institute, where he has taught since earning his Ph.D. at the University of Michigan in 1968. He has written over 50 research papers in optimization and operations research as well as two textbooks. He was a visiting professor in 1975–1976 at the Center for Operations Research and Econometrics in Belgium, and in 1982–1983 at Ecole Polytechnic in Lausanne, Switzerland. For the past five years, Joe has been active in innovative uses of technology in the calculus and workshop calculus courses.

New Directions in Elementary Differential Equations

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsWilliam E. BoyceWilliam E. Boyce has his B.A. from Rhodes College and his M.S. and Ph.D. (1955) from Carnegie-Mellon University, where his dissertation director was George Handelman. He now occupies the Edward P. Hamilton Chair of Science Education at Rensselaer Polytechnic Institute. Research interests include random and deterministic boundary value problems and their applications in continuum mechanics. With the late R. C. DiPrima he wrote Elementary Differential Equations and Boundary Value Problems, now in its fifth edition. He enjoys the Adirondack Mountains and following the fortunes of the St. Louis Cardinals. His wife, Elsa K. Boyce, is also a mathematician; they have three children and one grandchild.

Asymptotic Approximation of the Solution of a Random Boundary Value Problem Containing Small White Noise

This paper considers a nonlinear random differential equation [formula] where α(ω) is F1-measurable and w is an Rm-valued Wiener process. By introducing a weak problem, the shooting method can be used to prove the uniqueness of the Rn-valued Ft-measurable solution x(t) in the meaning of large probability. If the parameter ϵ is small, then x(t) = x0(t) + ϵx1(t) + O(ϵ2), where x0(t) is the solution with ϵ = 0 and x1(t) satisfies a linear random boundary value problem. For simplicity the discussion is given in the scalar case, but extensions to higher dimensions are readily apparent.

Almost sure convergence of a finite element approximations for the random Sturm-Liouville boundary value problem

This is a continuation of our previous work [3]. Convergent properties of the finite element approximations and a sample mean for the random Sturm-Liouville boundary value problem is discussed.

Nonlinear random elliptic boundary value problem

In this paper we obtain a solvability result for random elliptic boundary value problem. Our result extends Browder's [2] main theorem to the random case. We use author's [4] fixed point theorem for this purpose.