Hand Side Function Research Articles

Periodic solutions for nonlinear systems of Ode's with generalized variable exponents operators

Ordinary and partial differential equations with operators containing the p-Laplace operator, the so called ϕ-Laplace operator, operators with variable exponents and the double phase operators have been studied thoroughly for at least the last 20 years and the amount of papers dealing with these subjects is huge. In this paper we deal with the problem of the existence of periodic solutions to some system of ODE's involving a fairly general operator generalizing in this form corresponding results for the areas of research just mentioned. In our results neither the function generating the differential operator nor the right hand side function in the boundary value problem considered are necessarily equal to the gradient of a function. Our main result is a here is a continuation theorem based on the Leray-Schauder degree theory which allows us to obtain the existence of solutions for the boundary value problem considered.

Second order estimates for convex solutions of degenerate k-Hessian equations

The C1,1 estimate of the Dirichlet problem for degenerate k-Hessian equations with non-homogenous boundary conditions is an open problem, if the right hand side function f is only assumed to satisfy f1/(k−1)∈C1,1. In this paper, we solve this problem for convex solutions defined in the strictly convex bounded domain.

NUMERICAL METHOD FOR SOLVING THE DIRICHLET BOUNDARY VALUE PROBLEM FOR NONLINEAR TRIHARMONIC EQUATION

In this work, we consider the Dirichlet boundary value problem for nonlinear triharmonic equation. Due to the reduction of the problem to operator equation for the pair of the right hand side function and the unknown second normal derivative of the function to be sought, we design an iterative method at both continuous and discrete levels for numerical solution of the problem. Some examples demonstrate that the numerical method is of fourth order convergence. When the right hand side function does not depend on the unknown function and its derivatives, the numerical method gives more accurate results in comparison with the results obtained by the interior method of Gudi and Neilan.

Parameter Estimation for Dynamical Systems Using a Deep Neural Network

The deep neural network (DNN) was applied for estimating a set of unknown parameters of a dynamical system whose measured data are given for a set of discrete time points. We developed a new vectorized algorithm that takes the number of unknowns (state variables) and number of parameters into consideration. The algorithm, first, trains the network to determine weights and biases. Next, the algorithm solves the systems of algebraic equations to estimate the parameters of the system. If the right hand side function of the system is smooth and the system have equal numbers of unknowns and parameters, the algorithm solves the algebraic equation at the discrete point where absolute error between the neural network solutions and the measured data is minimum. This improves the accuracy and reduces computational time. Several tests were carried out in linear and non-linear dynamical systems. Last, we showed that the DNN approach is more successful in terms of computational time as the number of hidden layers increases.

An interior a priori estimate for solutions to Monge-Ampère equations with right-hand side close to a positive constant

We consider Monge-Ampère equations with the right hand side function close to a positive constant and from a function class that is larger than any Hölder class and smaller than the Dini-continuous class. We establish an upper bound for the modulus of continuity of the solution's second derivatives. This bound depends exponentially on a quantity similar to but larger than the Dini semi-norm. We establish explicit bounds regarding the shape of the sequence of shrinking sections, hence revealing the nature of such exponential dependence.

Regularity of the solution to fractional diffusion, advection, reaction equations in weighted Sobolev spaces

In this article we investigate the regularity of the solution to the fractional diffusion, advection, reaction equation on a bounded domain in R1. The analysis is performed in the weighted Sobolev spaces, H(a,b)s(I). Three different characterizations of H(a,b)s(I) are presented, together with needed embedding theorems for these spaces. The analysis shows that the regularity of the solution is bounded by the endpoint behavior of the solution, which is determined by the parameters α and r defining the fractional diffusion operator. Additionally, the analysis shows that for a sufficiently smooth right hand side function, the regularity of the solution to fractional diffusion reaction equation is lower than that of the fractional diffusion equation. Also, the regularity of the solution to fractional diffusion advection reaction equation is two orders lower than that of the fractional diffusion reaction equation.

Some estimates of virtual element methods for fourth order problems

<p style='text-indent:20px;'>In this paper, we employ the techniques developed for second order operators to obtain the new estimates of Virtual Element Method for fourth order operators. The analysis bases on elements with proper shape regularity. Estimates for projection and interpolation operators are derived. Also, the biharmonic problem is solved by Virtual Element Method, optimal error estimates were obtained. Our choice of the discrete form for the right hand side function relaxes the regularity requirement in previous work and the error estimates between exact solutions and the computable numerical solutions were proved.</p>

An asymptotic preserving semi-implicit multiderivative solver

In this work we construct a multiderivative implicit-explicit (IMEX) scheme for a class of stiff ordinary differential equations (ODEs). Our solver is high-order accurate and has an asymptotic preserving (AP) property for a large class of singularly perturbed ODEs. In this context, the AP property means that the singular limit is discretely preserved when a stiff parameter ε goes to zero. The proposed method is based upon a two-derivative backward Taylor series base solver, which we show has an AP property. Higher order accuracies are found by iterating the result over a high-order multiderivative interpolant of the right hand side function, which we again prove has an AP property. Theoretical results showcasing the asymptotic consistency as well as the high-order accuracy of the solver are presented. In addition, an extension of the solver to an arbitrarily split right hand side function is also offered. Numerical results for a collection of standard test cases from the literature are presented that support the theoretical findings of the paper.

A note on primal-dual stability in infinite linear programming

In this note we analyze the simultaneous preservation of the consistency (and of the inconsistency) of linear programming problems posed in infinite dimensional Banach spaces, and their corresponding dual problems, under sufficiently small perturbations of the data. We consider seven different scenarios associated with the different possibilities of perturbations of the data (the objective functional, the constraint functionals, and the right hand-side function), i.e., which of them are known, and remain fixed, and which ones can be perturbed because of their uncertainty. The obtained results allow us to give sufficient and necessary conditions for the coincidence of the optimal values of both problems and for the stability of the duality gap under the same type of perturbations. There appear substantial differences with the finite dimensional case due to the distinct topological properties of cones in finite and infinite dimensional Banach spaces.

AN INVERSE PROBLEM FOR THE PSEUDO-PARABOLIC EQUATION FOR A STURM-LIOUVILLE OPERATOR

A class of inverse problems for restoring the right-hand side of the pseudo-parabolic equation for Sturm-Liouville operator is considered. The inverse problem is to be well-posed in the sense of Hadamard whenever an overdetermination condition of the final temperature is given. Mathematical statements involve inverse problems for the pseudo-parabolic equation in which, solving the equation, we have to find the unknown right-hand side depending only on the space variable. We prove the existence and uniqueness of classical solutions to the problem. The proof of the existence and uniqueness results of the solutions is carried out by using L-Fourier analysis. The mentioned results are presented as well as for the fractional time pseudo-parabolic equation.Inverse problem of identifying the right hand side function of pseudo-parabolic equation from the local overdetermination condition, which has important applications in various areas of applied science and engineering, alsosuch problems are modeled using common homogeneous left-invariant hypoelliptic operators on common graded Lie groups.

On the convergence of spectral deferred correction methods

In this work we analyze the convergence properties of the Spectral Deferred Correction (SDC) method originally proposed by Dutt et al. [BIT, 40 (2000), pp. 241--266]. The framework for this high-order ordinary differential equation (ODE) solver is typically described wherein a low-order approximation (such as forward or backward Euler) is lifted to higher order accuracy by applying the same low-order method to an error equation and then adding in the resulting defect to correct the solution. Our focus is not on solving the error equation to increase the order of accuracy, but on rewriting the solver as an iterative Picard integral equation solver. In doing so, our chief finding is that it is not the low-order solver that picks up the order of accuracy with each correction, but it is the underlying quadrature rule of the right hand side function that is solely responsible for picking up additional orders of accuracy. Our proofs point to a total of three sources of errors that SDC methods carry: the error at the current time point, the error from the previous iterate, and the numerical integration error that comes from the total number of quadrature nodes used for integration. The second of these two sources of errors is what separates SDC methods from Picard integral equation methods; our findings indicate that as long as difference between the current and previous iterate always gets multiplied by at least a constant multiple of the time step size, then high-order accuracy can be found even if the underlying "solver" is inconsistent the underlying ODE. From this vantage, we solidify the prospects of extending spectral deferred correction methods to a larger class of solvers to which we present some examples.

Предсказание эволюции динамических систем остаточными нейронными сетями

Forecasting time series and time-dependent data is a common problem in many applications. One typical example is solving ordinary differential equation (ODE) systems $\dot{x}=F(x)$. Oftentimes the right hand side function $F(x)$ is not known explicitly and the ODE system is described by solution samples taken at some time points. Hence, ODE solvers cannot be used. In this paper, a data-driven approach to learning the evolution of dynamical systems is considered. We show how by training neural networks with ResNet-like architecture on the solution samples, models can be developed to predict the ODE system solution further in time. By evaluating the proposed approaches on three test ODE systems, we demonstrate that the neural network models are able to reproduce the main dynamics of the systems qualitatively well. Moreover, the predicted solution remains stable for much longer times than for other currently known models.

A simplex-type algorithm for continuous linear programs with constant coefficients

We consider continuous linear programs over a continuous finite time horizon $T$, with a constant coefficient matrix, linear right hand side functions and linear cost coefficient functions, where we search for optimal solutions in the space of measures or of functions of bounded variation. These models generalize the separated continuous linear programming models and their various duals, as formulated in the past by Anderson, by Pullan, and by Weiss. In previous papers we have shown that these problems possess optimal strongly dual solutions. We also have presented a detailed description of optimal solutions and have defined a combinatorial analogue to basic solutions of standard LP. In this paper we present an algorithm which solves this class of problems in a finite bounded number of steps, using an analogue of the simplex method, in the space of measures.

Least-Squared Mixed Variational Formulation Based on Space Decomposition for a Kind of Variable-Coefficient Fractional Diffusion Problems

In this paper, we decompose the fractional derivative space as the direct-sum of a fractional Sobolev space and a singular space spanned by $$x^{-\beta }$$ and then propose a $$x^{-\beta }$$ -independent mixed type variational formulation over the commonly used Sobolev spaces for a kind of variable-coefficient fractional diffusion equations, based on the least-squared techniques and the merits of the direct-sum decomposition. We then prove the existence and uniqueness of the variational formulation, show the equivalence between the variational formulation and the fractional diffusion equation and discuss the regularity of the solution to the equation with a general right hand side function. As a consequence, an easily-computed and optimal-order-convergent least-squared mixed finite element method is established. The optimal-order numerical analysis with supporting numerical experiments is also conducted.

Error propagation for implicit–explicit general linear methods

We consider the class of implicit–explicit general linear methods (IMEX). Such schemes are designed for ordinary differential equation systems with right hand side function splitted into stiff and non-stiff parts. We investigate error propagation of IMEX methods up to the terms of order p+2. In addition, we construct IMEX schemes of order p and stage order q=p, p≤4 and we verify the performance of methods in several numerical experiments.

Probabilistic evolution theory for explicit autonomous ordinary differential equations: recursion of squarified telescope matrices and optimal space extension

Probabilistic evolution theory facilitates the solution of initial value problem of explicit autonomous ordinary differential equations with second degree multinomial right hand side functions. Its formulation has components we call telescope matrices. The matrices grow in size very rapidly and has many zeroes and repeating structures. In order to avoid the computational complexity coming from telescope matrices, squarified telescope matrices are utilized. Their calculation is through a recursion. This recursion has been used in several works by the authors and their colleagues but its proof was not given. This work gives the proof of the recursion and all the surrounding details. A second purpose of this work is to provide a method for most facilitative (optimal) space extension. Space extension is needed for using probabilistic evolution theory when degree of multinomiality of the right hand side functions is more than two. For this purpose, an approach using method of exhaustion (brute-force) is proposed.

Probabilistic evolution theory for the solution of explicit autonomous ordinary differential equations: squarified telescope matrices

Probabilistic evolution theory (PREVTH) is used for the solution of initial value problems of first order explicit autonomous ordinary differential equation sets with second degree multinomial right hand side functions. It is an approximation method based on Kronecker power series: a rewriting of multivariate Taylor series using matrices having certain flexible parameters. Kronecker power series have matrices which are called telescope matrices: \(n \times n^{j+1}\) matrices where j is the index of summation. The additive terms of each telescope matrix is formed through Kronecker product from both sides by Kronecker powers of identity matrices. Recently, squarification is proposed in order to avoid the growing of the matrices in size at each additive term of the series. This paper explains the squarification procedure: the procedure used in order to avoid Kronecker multiplications within PREVTH so that the sizes of the matrices do not grow and so that the amount of necessary computation is reduced. The recursion between squarified matrices is also given. As a numerical application, the solution of a Henon–Heiles system is provided.

Analytical Jacobian-vector products for the matrix-free time integration of partial differential equations

Many scientific and engineering applications require the solution of large systems of initial value problems arising from method of lines discretization of partial differential equations. For systems with widely varying time scales, or with complex physical dynamics, implicit time integration schemes are preferred due to their superior stability properties. These schemes solve at each step linear systems with matrices formed using the Jacobian of the right hand side function. For large applications iterative linear algebra methods, which make use of Jacobian-vector products, are employed. This paper studies the impact that the method of computing Jacobian-vector products has on the overall performance and accuracy of the time integration process. The analysis shows that the most beneficial approach is the direct computation of exact Jacobian-vector products in the context of matrix-free time integrators. This approach does not suffer from approximation errors, reuses the parallelism and data distribution already present in the right-hand side vector computations, and avoids storing or operating on the entire Jacobian matrix.

$C^\infty$ Local solutions of elliptical $2-$Hessian equation in $\mathbb{R}^3$

In this work, we study the existence of $C^{\infty}$ local solutions to $2$-Hessian equation in $\mathbb{R}^{3}$. We consider the case that the right hand side function $f$ possibly vanishes, changes the sign, is positively or negatively defined. We also give the convexities of solutions which are related with the annulation or the sign of right-hand side function $f$. The associated linearized operator are uniformly elliptic.

On Solvability of the Neumann Boundary Value Problem for Non-homogeneous Biharmonic Equation

In this work we investigate the Neumann boundary value problem in the unit ball for a nonhomogeneous biharmonic equation. It is well known, that even for the Poisson equation this problem does not have a solution for an arbitrary smooth right hand side and boundary functions; it follows from the Green formula, that these given functions should satisfy a condition called the solvability condition. In the present paper these solvability conditions are found in an explicit form for the natural generalization of the Neumann problem for the non-homogeneous biharmonic equation. The method used is new for these type of problems. We first reduce this problem to the Dirichlet problem, then use the Green function of the Dirichlet problem recently found by T. Sh. Kal’menov and D. Suragan