Value Problems Of Second Order Research Articles

Economical handling of Runge–Kutta–Nyström step rejection

Runge–Kutta–Nyström (RKN) pairs are amongst the most popular methods for solving numerically Initial Value Problems of second order. Here we consider the special case with first derivative absent from the right hand side of the classical definition of this problem. While using a RKN pair, we may experience rejection of some steps through the interval of integration. Traditionally, the whole amount of evaluations are then dropped and we proceed with a completely new round of computations. In this work we propose avoiding this waste and continue by using again the rejected RKN stages. We especially focus on a RKN pair of orders six and four. After a step rejection we use again all the previously evaluated stages and add only one new stage. We proceed by evaluating the output in a smaller step. By this technique we manage to reduce significantly the cost in a set of problems that are known to pose difficulties in RKN step size changing algorithms.

Hybrid Hybrid one step trigonometrically scheme with oscillatory solutions for solving initial value problems of second order

A hybrid one step scheme with coefficients based on the step size and frequency of the trigonometric function used as basis function is derived. The constructed scheme is use to solve some problems with oscillating solutions. Stability properties of the developed scheme was also discussed. Numerical result obtained through the implementation of the scheme show its computation efficiency and accuracy over some of the existing ones (Abhulimen and Okunuga[1], Adeniran, Odejide and Ogundare [2], Alabi, Oladipo and Adesanya[3], Awoyemi, Adebile, Adesanya, and Anake, [5]) in the literature.

Nonlinear conservation laws for the Schr\xf6dinger boundary value problems of second order

In this paper, we apply a reliable combination of maximum modulus method with respect to the Schrödinger operator and Phragmén–Lindelöf method to investigate nonlinear conservation laws for the Schrödinger boundary value problems of second order. As an application, we prove the global existence to the solution for the Cauchy problem of the semilinear Schrödinger equation. The results reveal that this method is effective and simple.

Spectral analysis of hypoelliptic Višik–Ventcel’ boundary value problems

This paper is devoted to the study of a class of hypoelliptic Visik–Ventcel’ boundary value problems for second order, uniformly elliptic differential operators. Our boundary conditions are supposed to correspond to the diffusion phenomenon along the boundary, the absorption and reflection phenomena at the boundary in probability. If the absorbing boundary portion is not a trap for Markovian particles, then we can prove two existence and uniqueness theorems of the non-homogeneous Visik–Ventcel’ boundary value problem in the framework of $$L^{2}$$ Sobolev spaces. Moreover, if the absorbing boundary portion is empty, then we can prove a generation theorem of analytic semigroups for the closed realization of the uniformly elliptic differential operator associated with the hypoelliptic Visik–Ventcel’ boundary condition in the $$L^{2}$$ topology. As a by-product, this paper is the first time to prove the angular distribution of eigenvalues, the asymptotic eigenvalue distribution and the completeness of generalized eigenfunctions of the closed realization, similar to the elliptic (non-degenerate) case.

Homotopy Analysis Method for Solving Initial Value Problems of Second Order with Discontinuities

In this paper, the standard homotopy analysis method was applied to initial value problems of the second order with some types of discontinuities, for both linear and nonlinear cases. To show the high accuracy of the solution results compared with the exact solution, a comparison of the numerical results was made applying the standard homotopy analysis method with the iteration of the integral equation and the numerical solution with the Simpson rule. Also, the maximum absolute error, , the maximum relative error, the maximum residual error and the estimated order of convergence were given. The research is meaningful and I recommend it to be published in the journal.

On existence and uniqueness of viscosity solutions for second order fully nonlinear PDEs with Caputo time fractional derivatives

Initial-boundary value problems for second order fully nonlinear PDEs with Caputo time fractional derivatives of order less than one are considered in the framework of viscosity solution theory. Associated boundary conditions are Dirichlet and Neumann, and they are considered in the strong sense and the viscosity sense, respectively. By a comparison principle and Perron’s method, unique existence for the Cauchy–Dirichlet and Cauchy–Neumann problems are proved.

Avoiding the order reduction when solving second-order in time PDEs with Fractional Step Runge–Kutta–Nyström methods

We study some of the main features of Fractional Step Runge–Kutta–Nyström methods when they are used to integrate Initial–Boundary Value Problems of second order in time, in combination with a suitable spatial discretization. We focus our attention on the order reduction phenomenon, which appears if classical boundary conditions are taken at the internal stages. This drawback is specially hard when time dependent boundary conditions are considered. In this paper we present an efficient technique, very simple and computationally cheap, which allows us to avoid the order reduction; such technique consists in modifying the boundary conditions for the internal stages of the method.

A Combined Method For The Numerical Solution Of Boundary Value Problems Of Second Order

In this work, a technique based on combination between collocation and spline method along with the shooting method is proposed for the solution of boundary value problem of order 2. Numerical results show that the method is simple and effective.

MIXED BOUNDARY VALUE PROBLEMS FOR SECOND ORDER DIFFERENTIAL EQUATIONS WITH DIFFERENT DEVIATED ARGUMENTS

This paper deals with second order differential equations with different deviated arguments (t) and (t, (t)). We investigate the existence of solutions of such problems with nonlinear mixed boundary conditions. To obtain corresponding results we apply the monotone iterative technique and the lower-upper solutions method. Two examples demonstrate the application of our results.

Symmetry of Solutions to Variational Problems for Nonlinear Elliptic Equations via Reflection Methods

We discuss some recent results on symmetry of solutions of nonlinear partial differential equations. We focus on elliptic and degenerate elliptic boundary value problems of second order with variational structure and the simple looking case where the underlying domain is radially symmetric. In this setting, we study solutions which are given as minimizers of constrained minimization problems or have low Morse index, and we examine which amount of symmetry of the data is inherited by these solutions. We highlight how the answer to this general question depends on specific assumptions on the data. The underlying techniques collected in this survey are elementary as they solely rely on hyperplane reflections and well known analytical and topological tools, but they yield surprisingly general results in situations where classical methods do not apply.

BOUNDARY VALUE PROBLEMS FOR SECOND ORDER CONVEX AND NONCONVEX DIFFERENTIAL INCLUSIONS WITH INTEGRAL BOUNDARY CONDITIONS

BOUNDARY VALUE PROBLEMS FOR SECOND ORDER CONVEX AND NONCONVEX DIFFERENTIAL INCLUSIONS WITH INTEGRAL BOUNDARY CONDITIONS

An iterative substructuring method for the hp‐version of the BEM on quasi‐uniform triangular meshes

AbstractWe study an additive Schwarz based preconditioner for the hp‐version of the boundary element method with quasi‐uniform triangular meshes and for hypersingular integral operators. The model problem is Laplace's equation exterior to an open surface and is generic for elliptic boundary value problems of second order in bounded and unbounded domains with closed or open boundary. The preconditioner is based on a nonoverlapping subspace decomposition into a so‐called wire basket space and interior functions for each element. We prove that the condition number of the preconditioned stiffness matrix has a bound, which is independent of the mesh size h and which grows only polylogarithmically in p, the maximum polynomial degree. Numerical experiments confirm this result. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 879–903, 2007

On boundary value problems of second order perturbed dynamic inclusions on time scales

In this paper, an existence criterion is established for a class of second order perturbed dynamic inclusions on time scales with boundary conditions. The main approach is a fixed-point theorem for multi-valued maps due to Dhage.

On the boundary value problems for ordinary differential equations with isolated singularity

We consider some two-point boundary value problems for second order o.d.e. with strong singularity. The asymptotics of the solutions at the singular point are given. The existence and uniqueness of the solutions of those problems are prooved.

Nonlinear Boundary Value Problems for Second Order Differential Inclusions

In this paper we study two boundary value problems for second order strongly nonlinear differential inclusions involving a maximal monotone term. The first is a vector problem with Dirichlet boundary conditions and a nonlinear differential operator of the form x ↦ a(x, x′)′. In this problem the maximal monotone term is required to be defined everywhere in the state space ℝN. The second problem is a scalar problem with periodic boundary conditions and a differential operator of the form x ↦ (a(x)x′)′. In this case the maximal monotone term need not be defined everywhere, incorporating into our framework differential variational inequalities. Using techniques from multivalued analysis and from nonlinear analysis, we prove the existence of solutions for both problems under convexity and nonconvexity conditions on the multivalued right-hand side.

<![CDATA[<B>Numerical treatment of boundary value problems for second order singularly perturbed delay differential equations</B>]]>

In this paper, the boundary value problems for second order singularly perturbed delay differential equations are treated. A generic numerical approach based on finite difference is presented to solve such boundary value problems. The stability and convergence analysis of the method is studied. The solution of the boundary value problems when delay is zero, exhibits layer behavior. Here, the study focuses on the effect of delay on the boundary layer behavior of the solution via numerical approach. The effect of the delay on the boundary layer behavior of the solution is shown by carrying out some numerical experiments.

Linear Elliptic Boundary Value Problems with Non-Smooth Data: Campanato Spaces of Functionals

In this paper linear elliptic boundary value problems of second order with non-smooth data L∞-coefficients, sets with Lipschitz boundary, regular sets, non-homogeneous mixed boundary conditions) are considered. It will be shown that such boundary value problems generate isomorphisms between certain Sobolev-Campanato spaces of functions and functionals, respectively.

Linear Elliptic Boundary Value Problems with Non - Smooth Data: Normal Solvability on Sobolev - Campanato Spaces

In this paper linear elliptic boundary value problems of second order with non – smooth data (L∞ – coeffcients, Lipschitz domains, regular sets, non – Chomogeneous mixed boundary conditions) are considered. It is shown that such boundary value problems generate Fredholm operators between appropriate Sobolev – Campanato spaces, that the weak solutions are Hölder continuous up to the boundary and that they depend smoothly (in the sense of a Hölder norm) on the coefficients and on the right – hand sides of the equations and boundary conditions.

On the Solvability of Degenerate Quasilinear Parabolic Equations of Second Order

In this paper, we consider nonlinear degenerate quasilinear parabolic initial boundary value problems of second order. Using results from the theory of pseudomonotone operators, we show that there exists at least one weak solution in a suitable weighted Sobolev space.

On the Solvability of Degenerate Quasilinear Parabolic Equations of Second Order

In this paper, we consider a nonlinear degenerate quasilinear parabolic initial boundary value problems of second order. Using results from the theory of pseudomonotone operators, we show that there exists at least one weak solution in a suitable weighted Sobolev space.