Boundary Value Problems For Differential Equations Research Articles

Mesh-Clustered Gaussian Process Emulator for Partial Differential Equation Boundary Value Problems

Partial differential equations (PDEs) have become an essential tool for modeling complex physical systems. Such equations are typically solved numerically via mesh-based methods, such as finite element methods, with solutions over the spatial domain. However, obtaining these solutions are often prohibitively costly, limiting the feasibility of exploring parameters in PDEs. In this article, we propose an efficient emulator that simultaneously predicts the solutions over the spatial domain, with theoretical justification of its uncertainty quantification. The novelty of the proposed method lies in the incorporation of the mesh node coordinates into the statistical model. In particular, the proposed method segments the mesh nodes into multiple clusters via a Dirichlet process prior and fits Gaussian process models with the same hyperparameters in each of them. Most importantly, by revealing the underlying clustering structures, the proposed method can provide valuable insights into qualitative features of the resulting dynamics that can be used to guide further investigations. Real examples are demonstrated to show that our proposed method has smaller prediction errors than its main competitors, with competitive computation time, and identifies interesting clusters of mesh nodes that possess physical significance, such as satisfying boundary conditions. An R package for the proposed methodology is provided in an open repository.

Solving nonlinear ODEs with the ultraspherical spectral method

Abstract We extend the ultraspherical spectral method to solving nonlinear ordinary differential equation (ODE) boundary value problems. Naive ultraspherical Newton implementations usually form dense linear systems explicitly and solve these systems exactly by direct methods, thus suffering from the bottlenecks in both computational complexity and storage demands. Instead, we propose to use the inexact Newton–GMRES framework for which a cheap but effective preconditioner can be constructed and a fast Jacobian-vector multiplication can be effected, thanks to the structured operators of the ultraspherical spectral method. The proposed inexact Newton–GMRES–ultraspherical framework outperforms the naive implementations in both speed and storage, particularly for large-scale problems or problems whose linearization has solution-dependent variable coefficients in higher-order terms. Additional acceleration can be gained when the method is implemented with mixed precision arithmetic.

Spectral Analysis of One Class of Positive-Definite Operators and Their Application in Fractional Calculus

This paper is devoted to the spectral analysis of one class of integral operators, associated with the boundary-value problems for differential equations of fractional order. In particular, we show the positive definiteness of studying operators, which makes it possible to select areas in the complex plane where there are no eigenvalues for these operators.

Design of Sturm global attractors 1: Meanders with three noses, and reversibility.

We systematically explore a simple class of global attractors, called Sturm due to nodal properties, for the semilinear scalar parabolic partial differential equation (PDE) ut=uxx+f(x,u,ux) on the unit interval 0<x<1, under Neumann boundary conditions. This models the interplay of reaction, advection, and diffusion. Our classification is based on the Sturm meanders, which arise from a shooting approach to the ordinary differential equation boundary value problem of equilibrium solutions ut=0. Specifically, we address meanders with only three "noses," each of which is innermost to a nested family of upper or lower meander arcs. The Chafee-Infante paradigm, with cubic nonlinearity f=f(u), features just two noses. Our results on the gradient-like global PDE dynamics include a precise description of the connection graphs. The edges denote PDE heteroclinic orbits v1⇝v2 between equilibrium vertices v1,v2 of adjacent Morse index. The global attractor turns out to be a ball of dimension d, given as the closure of the unstable manifold Wu(O) of the unique equilibrium with maximal Morse index d. Surprisingly, for parabolic PDEs based on irreversible diffusion, the connection graph indicates time reversibility on the (d-1)-sphere boundary of the global attractor.

Existence of positive solutions for singular p-Laplacian Hadamard fractional differential equations with the derivative term contained in the nonlinear term

In this paper, based on the properties of Green function and the eigenvalue of a corresponding linear operator, the existence of positive solutions is investigated by spectral analysis for a infinite-points singular p-Laplacian Hadamard fractional differential equation boundary value problem, and an example is given to demonstrate the validity of our main results.

The existence of a solution for nonlinear fractional differential equations where nonlinear term depends on the fractional and first order derivative of an unknown function

In this paper, we consider the existence of solutions of the nonlinear fractional differential equation boundary-value problem D?* u(t) = f (t, u(t), u?(t), CD?u(t)), 0 &lt; t &lt; 1, 1 &lt; ? &lt; 2, 0 &lt; ? ? 1, u(0) = A, u(1) = Bu(?), where 0 &lt; ? &lt; 1, A ? 0, B? &gt; 1, D?* is the modified Caputo fractional derivative of order ?, CD? is the Caputo fractional derivative of order ?, and f is a function in C([0, 1] ? R ? R ? R). Existence results for a solution are obtained. Two examples are presented to illustrate the results.

A Computational Method for Solving a Boundary-Value Problem for Differential Equations with Piecewise Constant Argument of Generalized Type

A Computational Method for Solving a Boundary-Value Problem for Differential Equations with Piecewise Constant Argument of Generalized Type

Boundary-Value Problems for Differential-Difference Equations with Incommensurable Shifts of Arguments Reducible to Nonlocal Problems

We consider boundary-value problems for differential-difference equations containing incommensurable shifts of arguments in higher-order terms. We prove that in the case of finite orbits of boundary points generated by the set of shifts of the difference operator, the original problem is reduced to a boundary-value problem for differential equation with nonlocal boundary conditions.

Computation of reactive thermosolutal micropolar nanofluid Sakiadis convection flow with gold/silver metallic nanoparticles

In the present study, a mathematical model is developed by combining the Tiwari-Das nanofluid formulation with the Eringen micro-morphic model to simulate the thermo-solutal natural convection chemically reacting micropolar nanofluid flow from a permeable stretching surface with non-uniform heat source/sink effects. The transformed ordinary differential equation boundary value problem features linear momentum, angular momentum, energy and species conservation boundary layer equations with appropriate boundary conditions. This ninth order nonlinear system is solved with Runge-Kutta 45 Fehlberg method. The influence of the effect of the emerging parameters on the flow characteristics are visualized and tabulated. It is observed that increasing volume fraction decreases velocity whereas it elevates microrotation, temperature and nanoparticle concentration. Nanoparticle concentration are elevated for stronger destructive chemical reaction effect whereas they are suppressed with constructive chemical reaction. With greater micropolar boundary condition parameter, the velocity is elevated, microrotation but reduces temperature and thermal boundary layer thickness. Increasing nanoparticle volume fraction enhances both skin friction and couple stress but marginally reduces the Nusselt number. Finally, Au-water micropolar nanofluids achieve the highest skin friction and couple stress magnitudes, then Ag-water and finally Cu-water. Validation of solutions with earlier non-reactive studies in the absence of nanoparticle mass transfer are included.

Self-similar solutions for resistive diffusion, Ohmic heating, and Ettingshausen effects in plasmas of arbitrary β

Magneto-inertial fusion (MIF) approaches, such as the MagLIF experiment, use magnetic fields in dense plasma to suppress cross-field thermal conduction, attempting to reduce heat loss and trap alpha particles to achieve ignition. However, the magnetic field can introduce other transport effects, some of which are deleterious. An understanding of these processes is thus crucial for accurate modeling of MIF. We generalize past work exploiting self-similar solutions to describe transport processes in planar geometry and compare the model to the radiation-magnetohydrodynamics (MHDs) code Chimera. We solve the 1D extended MHD equations under pressure balance, making no assumptions about the ratio of magnetic and thermal pressures in the plasma. The resulting ordinary differential equation (ODE) boundary value problem is solved using a shooting method, combining an implicit ODE solver and a Newton–Raphson root finder. We show that the Nernst effect dominates over resistive diffusion in high β plasma, but its significance is reduced as the β decreases. On the other hand, we find that Ettingshausen and Ohmic heating effects are dominant in low β plasma and can be observable in even order unity β plasma, though in the presence of a strong temperature gradient heat conduction remains dominant. We then present a test problem for the Ohmic heating and Ettingshausen effects which will be useful to validate codes modeling these effects. We also observe that the Ettingshausen effect plays a role in preventing temperature separation when Ohmic heating is strong. Neglecting this term may lead to overestimates for the electron temperature at a vacuum–plasma interface, such as at the edge of a z-pinch. The model developed can be used to provide test problems with arbitrary boundary conditions for magnetohydrodynamics codes with the ability to freely switch on terms to compare their individual implementations.

USER-ORIENTED INTELLIGENCE MINING UNDER THE EXISTENCE OF SOLUTIONS TO INTEGRAL BOUNDARY VALUE PROBLEMS FOR FUZZY PARTIAL FRACTIONAL DIFFERENTIAL EQUATIONS

In order to accurately perceive user intention and improve the reliability of user intention information mining, the existence of solution of integral boundary value problem of fuzzy partial fractional differential equation is used to mine the information of user intention. Firstly, the periodic boundary value problem of fractional order fuzzy linear differential equation, periodic boundary value problem of fractional order fuzzy nonlinear differential equation and periodic boundary value problem of fractional order fuzzy coupled differential system are described in this research. The existence of the solution of the integral boundary value problem of the fuzzy partial fractional order differential equation is proved. Then a user-oriented information mining framework and an evaluation model based on the existence of solutions for the integral boundary value problem of fuzzy partial fractional differential equations are constructed. Finally, this research makes a case study and a comprehensive evaluation of user-oriented information mining based on the existence of solutions of integral boundary value problems for fuzzy partial fractional differential equations. The results show that the method based on the existence of the solution of the integral boundary value problem of fuzzy partial fractional order differential equation is feasible and scientific for the information mining of user intention. It is also concluded that this method is suitable for the modelling and solving of intelligence mining with complicated and unclear user intention. The research provides a good guidance for information mining oriented to user intention.

EXISTENCE OF MULTIPLE POSITIVE SOLUTIONS FOR CAPUTO FRACTIONAL DIFFERENTIAL EQUATION WITH INFINITE-POINT BOUNDARY VALUE CONDITIONS

<abstract> In this paper, the existence of positive solutions for Caputo fractional differential equations boundary value problem with infinite points contained in the boundary value condition, and based on Green's function and its properties, the existence of multiple positive solutions are obtained by Avery-Peterson fixed point theorem. </abstract>

"AN UNIFIED DIFFERENCE METHOD FOR THE NUMERICAL SOLUTION OF THE TROESCH’S BOUNDARY VALUE PROBLEM"

"In this article, the unified difference and quasi qudratization method are developed and discussed to obtain an approximate numerical solution for the Troesch’s problem, a nonlinear differential equation and corresponding boundary value problems. The degree of accuracy in numerical solution by the proposed method is good and comparable to other existing methods in literature for the range of values of parameter in Troesch’s problem."

RESEARCH ON BAND GAP CALCULATION METHOD OF PERIODIC STRUCTURE BASED ON ARTIFICIAL SPRING MODEL

The energy method is widely used in structural dynamicsanalysis with its advantage of converting the boundary value problems fordifferential equation into the functional extreme value problem, and hasalso been introduced into periodic structure band gap computation in recentyears. However, it is difficult to construct the displacement function whenusing traditional energy methods (such as the Rayleigh-Ritz method) foranalysis because of the certain complexity in boundary conditions of theperiodic structure. Additionally, the wave number term is contained in thedisplacement function so that the mass and stiffness matrix need to berecomputed continuously in the process of calculating the band gap ofscanning wave number, which leads to a large amount of calculation. For thatreason, this paper improved the traditional energy method by introducingartificial spring model to simulate various boundary conditions includingperiodic boundaries so that boundary constraints could be transformed intothe elastic potential energy of artificial springs and only the periodicboundary elastic potential energy in energy distributions contains the wavenumber term, by which the corresponding stiffness matrix only needed to berecomputed in the scanning process of wave number and other mass andstiffness matrices need to be calculated only once and then significantlyreduced the computational burden. The research results show that the methodin this paper is accurate and reliable. The calculation efficiency of thismethod is advantageous compared with the traditional energy method. Theadvantage of calculation efficiency of this method is more obvious comparedwith the traditional energy method in the situation that the mass andstiffness matrix promote in dimension, or the scanning points of wave numberincrease. In addition, the artificial spring model is more flexible andconvenient to use, and can be further adopted to band gap analysis of morecomplex periodic composite structures.

Maximal and minimal iterative positive solutions for $ p $-Laplacian Hadamard fractional differential equations with the derivative term contained in the nonlinear term

&lt;abstract&gt;&lt;p&gt;In this paper, the maximal and minimal iterative positive solutions are investigated for a singular Hadamard fractional differential equation boundary value problem with a boundary condition involving values at infinite number of points. Green's function is deduced and some properties of Green's function are given. Based upon these properties, iterative schemes are established for approximating the maximal and minimal positive solutions.&lt;/p&gt;&lt;/abstract&gt;

Mathematical Model of a Cracked Pipeline Subjected to Sinusoidal Force Excitation

The present work introduces a mathematical model of a cracked pipeline conveying liquid. This model uses governing equation of Euler – Bernoulli beam theory for a pipe conveying liquid. The crack introduced in the model is represented by two identical torsional springs and a sinusoidal excitation force was applied to one of the pipe ends for the purpose of crack detection. The model is solved numerically using MATLAB code bvp4c to solve linear, ordinary fourth order differential equations (boundary value problem) for the detection of the crack position. To group the variables in the dimensionless form, Buckingham Pi-theorem was used. The effects of the dimensionless parameters on crack position were examined. The results show that the value of dimensionless parameters of stiffness at crack and support has considerable effect on crack position. It also shows that the force amplitude and fluid flow properties has no effect on crack position which demonstrates the capacity of using the present technique with small force magnitude to avoid stress-strain problems on the pipeline and any excitation frequency can be used.

THE NUMERICAL SOLUTION OF THREE POINT THIRD ORDER BOUNDARY VALUE PROBLEMS IN ODES

In the present article, third order differential equation and corresponding boundary value problems considered for the numerical solution. An improved finite difference method developed for the approximate numerical solution. The proposed method converges and O(h2), the order of convergence under appropriate condition established. In the numerical experiment, model problems were considered for the purpose of testing the performance of the proposed method. The numerical results generated in numerical experiment are in good agreement with the theoretical discussion and development of the proposed method.

Existence and Multiplicity of Positive Solutions for a Class of m-Point Fractional Differential Equations with p-Laplacian Operators

In this paper, by constructing the properties of Green function and fixed point theorem on cone, we study the existence of solutions for a class of m-point fractional differential equation boundary value problems with p-Laplacian operator, and prove the existence and multiplicity of positive solutions for boundary value problems by constructing superlinear functions.

Ulam–Hyers Stability Analysis of a Three-Point Boundary-Value Problem for Fractional Differential Equations

We study the problem of existence and uniqueness of the solution of a three-point boundary-value problem for a differential equation of fractional order. Further, we investigate various kinds of the Ulam stability, such as the Ulam–Hyers stability, the generalized Ulam–Hyers stability, the Ulam–Hyers–Rassias stability, and the generalized Ulam–Hyers–Rassias stability for the analyzed problem. We also present examples to explain our results.

On the solvability fractional of a boundary value problem with new fractional integral

This work presents new techniques for finding solutions of linear fractional differential equation boundary value problem when the derivation is conformable fractional of Caputo type, the first technique we will study the method that converts an initial value problem to an equivalent linear ordinary differential equation of second order. In order to find solution by an other technique we introduce a new definition of fractional integral as an inverse of the conformable fractional derivative of Caputo. Also, some examples are included to improve the validity and applicability of the techniques.