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In this paper, we are interested in finding a numerical solution of a linear BVP for a Fredholm IDE with a degenerate kernel subjected to impulsive actions. By Dzumabaev's parametrization method the original problem is reduced to a multipoint BVP for the system of Fredholm IDEs with additional parameters. For fixed parameters, the special Cauchy problem for the system of FIDEs on subintervals is obtained and by using a solution to this problem, a system of algebraic equations in parameters is constructed. An algorithm for solving the BVP and its computational implementation is developed. In the algorithm, Cauchy problems for ODEs and the calculation of definite integrals are the main auxiliary problems. By using various numerical methods for solving these auxiliary problems, the proposed algorithm can be implemented in different ways. The program codes were written to solve the problem and all calculations are performed on the Matlab 2018 software platform. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of periodic solutions of fractional differential equations with periodic impulses. The first part of the paper is devoted to the uniqueness, existence and asymptotic stability results for periodic solutions of impulsive fractional differential equations with varying lower limits for standard nonlinear cases as well as for cases of weak nonlinearities, equidistant and periodically shifted impulses. We also apply our result to an impulsive fractional Lorenz system. The second part extends the study to periodic impulsive fractional differential equations with fixed lower limit. We show that in general, there are no solutions with long periodic boundary value conditions for the case of bounded nonlinearities. [ABSTRACT FROM AUTHOR]

This paper presents a new numerical method for the solution of ordinary differential equations (ODE). The linear second-order equations considered herein are solved using operational matrices of Wang-Ball Polynomials. By the improvement of the operational matrix, the singularity of the ODE is removed, hence ensuring that a solution is obtained. In order to show the employability of the method, several problems were considered. The results indicate that the method is suitable to obtain accurate solutions. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence and uniqueness of solutions for a new class of multi-point boundary value problems of multi-term fractional differential equations by using standard fixed point theorems. We also demonstrate the application of the obtained results with the aid of examples. The paper concludes with the study of multi-term fractional integro-differential equations supplemented with multi-point boundary conditions. Our results are new and contribute significantly to the existing literature on the topic. [ABSTRACT FROM AUTHOR]

The key of the reproducing kernel method (RKM) to solve the initial boundary value problem is to construct the reproducing kernel meeting the homogenous initial boundary conditions of the considered problems. The usual method is that the initial boundary conditions must be homogeneous and put them into space. Another common method is to put homogeneous or non-homogeneous conditions directly into the operator. In addition, we give a new numerical method of RKM for dealing with initial boundary value problems, homogeneous conditions are put into space, and for nonhomogeneous conditions, we put them into operators. The focus of this paper is to further verify the reliability and accuracy of the latter two methods. Through solving three numerical examples of integral-differential equations and comparing with other methods, we find that the two methods are useful. [ABSTRACT FROM AUTHOR]

In this paper, third and fourth order of accuracy stable difference schemes for approximately solving multipoint nonlocal boundary value problems for hyperbolic equations with the Neumann boundary conditions are considered. Stability estimates for the solutions of these difference schemes are presented. Finite difference method is used to obtain numerical solutions. Numerical results of errors and CPU times are presented and are analyzed. [ABSTRACT FROM AUTHOR]

The aim of this paper is to consider the asymptotic behavior of boundary value problems in n-dimensional domains with periodically placed particles, with a general microscopic boundary condition on the particles and a p-Laplace diffusion operator on the interior, in the case in which the particles are of critical size. We consider the cases in which 1 < p < n, n ≥ 3. In fact, in contrast to previous results in the literature, we formulate the microscopic boundary condition in terms of a Robin type condition, involving a general maximal monotone graph, which also includes the case of microscopic Dirichlet boundary conditions. In this way we unify the treatment of apparently different formulations, which before were considered separately. We characterize the so called "strange term" in the homogenized problem for the case in which the particles are balls of critical size. Moreover, by studying an application in Chemical Engineering, we show that the critically sized particles lead to a more effective homogeneous reaction than noncritically sized particles. [ABSTRACT FROM AUTHOR]

In this paper we study the nonexistence of solutions, which are stable or stable outside a compact set, possibly unbounded and sign-changing, of some nonlinear elliptic equations with mixed boundary value conditions. The main methods used are the integral estimates and the monotonicity formula. [ABSTRACT FROM AUTHOR]

Existence, uniqueness and continuous dependence results together with maximum principles represent key tools in the analysis of lattice reaction-diffusion equations. In this paper, we study these questions in full generality by considering nonautonomous reaction functions, possibly nonsymmetric diffusion and continuous, discrete or mixed time. First, we prove the local existence and global uniqueness of bounded solutions, as well as the continuous dependence of solutions on the underlying time structure and on initial conditions. Next, we obtain the weak maximum principle which enables us to get the global existence of solutions. Finally, we provide the strong maximum principle which exhibits an interesting dependence on the time structure. Our results are illustrated by the autonomous Fisher and Nagumo lattice equations and a nonautonomous logistic population model with a variable carrying capacity. [ABSTRACT FROM AUTHOR]

In this paper, the synchronization for a kind of first order quasilinear hyperbolic system is taken into account. In this system, all the equations share the same positive wave speed. To realize the synchronization, a uniform constructive method is adopted, rather than an iteration process usually used in dealing with nonlinear systems. Furthermore, similar results on the exact boundary synchronization by groups can be obtained for a kind of first order quasilinear hyperbolic system of equations with different positive wave speeds by groups. [ABSTRACT FROM AUTHOR]

In this paper, a novel technique incorporated the homotopy analysis method (HAM) with Coiflets is developed to obtain highly accurate solutions of the Föppl-von Kármán equations for large bending deflection. The characteristic scale transformation is introduced to nondimensionalize the governing equations. The results are obtained for the transformed nondimensional equations, which are in very excellent agreement with analytical ones or numerical benchmarks performing good efficiency and validity. Besides, we notice the nonlinearity of the Föppl-von Kármán equations is closely connected with the load and length-width ratio of the plate. For the case of the plate suffering tremendous loads, the traditional linear theory does not work, while our Coiflets solutions are still very accurate. It is expected that our proposed approach not only keeps the outstanding merits of the HAM technique for handling strong nonlinearity, but also improves on the computational efficiency to a great extent. [ABSTRACT FROM AUTHOR]

In this paper, we study some type of equations which may model the behavior of species inhabiting in some habitat. For our purpose, using a priori bounded techniques, we obtain a positive solution to a family of non-local partial differential equations with non-homogeneous boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, several sufficient conditions for oscillation of all solutions of fourth order functional differential equations of neutral type of the form (r (t) (y (t) + p (t) y (t − τ)) ″ ) ″ + q (t) G (y (t − σ)) = 0 $$\begin{array}{} \displaystyle \bigl(r(t)(y(t)+p(t)y(t-\tau))''\bigr)''+q(t)G\bigl(y(t-\sigma)\bigr)=0 \end{array}$$ are studied under the assumption ∫ 0 ∞ t r (t) d t = ∞ $$\begin{array}{} \displaystyle \int\limits^{\infty}_{0}\frac{t}{r(t)}{\rm d} t =\infty \end{array}$$ [ABSTRACT FROM AUTHOR]

In this paper, a family of second derivative generalized Adams-type methods (SDGAMs) is proposed. Here, a boundary value approach to the numerical solution of stiff initial value problems (IVPs) by means of second derivative linear multistep formulae (SDLMF) is presented. Stability analysis shows that these methods with order p = 2k + 2 for all values of the step-length k ≥ 1 are all 0v,k-v-stable and Av,k-v-stable which must be used with (v, k - v)- boundary conditions. [ABSTRACT FROM AUTHOR]

We consider a vector-Laplace problem posed on a two-dimensional surface embedded in a three-dimensional domain, which results from the modeling of surface fluids based on exterior Cartesian differential operators. The main topic of this paper is the development and analysis of a finite element method for the discretization of this surface partial differential equation. We apply the trace finite element technique, in which finite element spaces on a background shape-regular tetrahedral mesh that is surface independent are used for discretization. In order to satisfy the constraint that the solution vector field is tangential to the surface we introduce a Lagrange multiplier. We show well-posedness of the resulting saddle point formulation. A discrete variant of this formulation is introduced which contains suitable stabilization terms and is based on trace finite element spaces. For this method we derive optimal discretization error bounds. Furthermore algebraic properties of the resulting discrete saddle point problem are studied. In particular an optimal Schur complement preconditioner is proposed. Results of a numerical experiment are included. [ABSTRACT FROM AUTHOR]

In this paper we treat null-controllability properties for the linear Kuramoto- Sivashinsky equation on a network with two types of boundary conditions. More precisely, the equation is considered on a star-shaped tree with Dirichlet and Neumann boundary conditions. By using the moment theory we can derive null-controllability properties with boundary controls acting on the external vertices of the tree. In particular, the controllability holds if the anti-diffusion parameter of the equation does not belong to a critical countable set of real numbers. We point out that the critical set for which the null-controllability fails differs from the first model to the second one. [ABSTRACT FROM AUTHOR]

This paper is concerned with the study of the uniform decay rates of the energy associated with mixed problems involving the wave equation with nonlinear localized damping. The domain is an unbounded open set of R2 with finite measure and has an unbounded smooth boundary Γ = ΓN ∪ ΓD such that ΓN ∩ ΓD ≠ ∅. On ΓD and ΓN we place the homogeneous Dirichlet and Neumann boundary conditions, respectively. Due to lack of local regularity, we used the elliptic decomposition of the solution as did Grisvard [J. Math. Pures Appl. (9), 68 (1989), pp. 215-259] combined with the use of appropriated cutoff functions. [ABSTRACT FROM AUTHOR]

The improved Boussinesq equation is studied in this paper. Control properties for this equation posed on a bounded interval are first considered. When the control acts through the Dirichlet boundary condition the linearized system is proved to be approximately but not spectrally controllable. In a second part, the equation is posed on the one-dimensional torus and distributed moving controls are considered. Under some condition on the velocity at which the control moves, exact controllability results for both linear and nonlinear improved Boussinesq equations are obtained applying the moment method and a fixed point argument. [ABSTRACT FROM AUTHOR]

In this paper, we study the inverse boundary value problem for the wave equation with a view towards an explicit reconstruction procedure. We consider both the anisotropic problem where the unknown is a general Riemannian metric smoothly varying in a domain and the isotropic problem where the metric is conformal to the Euclidean metric. Our objective in both cases is to construct the metric, using either the Neumann-to-Dirichlet (N-to-D) map or Dirichlet-to-Neumann (D-to-N) map as the data. In the anisotropic case we construct the metric in the boundary normal (or semigeodesic) coordinates via reconstruction of the wave field in the interior of the domain. In the isotropic case we can go further and construct the wave speed in the Euclidean coordinates via reconstruction of the coordinate transformation from the boundary normal coordinates to the Euclidean coordinates. Both cases utilize a variant of the Boundary Control method, and work by probing the interior using special boundary sources. We provide a computational experiment to demonstrate our procedure in the isotropic case with N-to-D data. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of solutions for fractional differential equations of arbitrary order with nonlocal integral boundary conditions. The existence results are obtained by applying Krasnoselskii's fixed point theorem and Leray-Schauder degree theory, while the uniqueness of the solutions is established by means of Banach's contraction mapping principle. The paper concludes with illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper, we study the quenching behavior for a one-dimensional quasilinear parabolic equation with singular reaction term and singular boundary flux. Under certain conditions on the initial data, we show that quenching occurs only on the boundary in finite time. Moreover, we derive some lower and upper bounds of the quenching rate and get some estimates for the quenching time. [ABSTRACT FROM AUTHOR]

We consider in this paper the mathematical and numerical modeling of reflective boundary conditions (BC) associated to Boltzmann–Poisson systems, including diffusive reflection in addition to specularity, in the context of electron transport in semiconductor device modeling at nano scales, and their implementation in Discontinuous Galerkin (DG) schemes. We study these BC on the physical boundaries of the device and develop a numerical approximation to model an insulating boundary condition, or equivalently, a pointwise zero flux mathematical condition for the electron transport equation. Such condition balances the incident and reflective momentum flux at the microscopic level, pointwise at the boundary, in the case of a more general mixed reflection with momentum dependant specularity probability p ( k → ) . We compare the computational prediction of physical observables given by the numerical implementation of these different reflection conditions in our DG scheme for BP models, and observe that the diffusive condition influences the kinetic moments over the whole domain in position space. [ABSTRACT FROM AUTHOR]

In this paper we study the isotropic realizability of a given non smooth gradient field ∇u defined in ℝd, namely when one can reconstruct an isotropic conductivity σ > 0 such that σ∇u is divergence free in ℝd. On the one hand, in the case where ∇u is non-vanishing, uniformly continuous in ℝd and Δu is a bounded function in ℝd, we prove the isotropic realizability of ∇u using the associated gradient flow combined with the DiPerna, Lions approach for solving ordinary differential equations in suitable Sobolev spaces. On the other hand, in the case where ∇u is piecewise regular, we prove roughly speaking that the isotropic realizability holds if and only if the normal derivatives of u on each side of the gradient discontinuity interfaces have the same sign. Some examples of conductivity reconstruction are given. [ABSTRACT FROM AUTHOR]

In this paper we review some results obtained for a distributed control problem regarding a class of phase field systems of Caginalp type with logarithmic potential. The aim of the control problem is forcing the location of the diffuse interface to be as close as possible to a prescribed set. However, due to some discontinuity in the cost functional, we have to regularize it and solve the related control problem for the approximation. We discuss the necessary optimality conditions. [ABSTRACT FROM AUTHOR]

A new symmetric six-step method with improved phase properties is introduced, for the first time in the literature, in this paper. The proposed method: (1) is a symmetric nonlinear six-step method, (2) is of three-stages, (3) is of twelfth algebraic order, (4) has vanished the phase-lag and (5) has vanished the derivatives of the phase-lag up to order five. For the new symmetric six-step method we present a detailed theoretical, numerical and computational analysis which consists of: (a) the development of the new three-stages symmetric six-step method, (b) the computation of the local truncation error of the new scheme, (c) the comparative error analysis of the new algorithm with other schemes of the same family: (i) the classical algorithm of the family (i.e. the method with constant coefficients), (ii) the recently proposed method of the same family with vanished phase-lag and its first derivative, (iii) the recently proposed method of the same family with vanished phase-lag and its first and second derivatives, (iv) the recently proposed method of the same family with vanished phase-lag and its first, second and third derivatives and finally (v) the recently proposed method of the same family with vanished phase-lag and its first, second, third and fourth derivatives, (d) the stability and the interval of periodicity analysis for the new developed method and finally (e) the study of the accuracy and computational effectiveness of the newproposed method for the solution of the Schrödinger equation. The theoretical, numerical and computational analysis of the newobtained three-stages symmetric six-step method show the efficiency of the new algorithm compared with other known or recently developed schemes of the literature. [ABSTRACT FROM AUTHOR]

We investigate a new kind of anti-periodic type boundary value problems of sequential fractional differential equation of order q ∊ (2; 3]. We make use of Banach's contraction mapping principle to obtain the uniqueness result while the existence of solutions is established via Krasnoselskii's fixed point theorem and Leray-Schauder nonlinear alternative. The paper concludes with some illustrative examples. [ABSTRACT FROM AUTHOR]

This paper presents a novel shooting method for solving two-point boundary value problems for second order ordinary differential equations. The method works as follows: first, a guess for the initial condition is made and an integration of the differential equation is performed to obtain an initial value problem solution; then, the end value of the solution is used in a simple iteration formula to correct the initial condition; the process is repeated until the second boundary condition is satisfied. The iteration formula is derived utilizing an auxiliary function that satisfies both boundary conditions and minimizes the H 1 semi-norm of the difference between itself and the initial value problem solution. [ABSTRACT FROM AUTHOR]

This paper presents sufficient conditions for the solvability of the third order three point boundary value problem The arguments apply Green's function associated to the linear problem and the Guo-Krasnosel'skiĭ theorem of compression-expansion cones. The dependence on the first derivatives is overcome by the construction of an adequate cone and suitable conditions of superlinearity/sublinearity near 0 and $$+\infty $$ . Last section contains an example to illustrate the applicability of the theorem. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to discuss numerical solutions of differential equations including the evolution, progress and types of differential equations. Special attention is given to the solution of differential equations by application of spline functions. Here we are interested in differential equation based problems and their solutions using polynomial and nonpolynomial splines of different orders. It contains crux of various recent research papers based on application of splines of different orders. [ABSTRACT FROM AUTHOR]

Abstract: Traditional approach to effectiveness factor calculations uses finite difference methods to discretize two-point boundary value differential equations. The finite difference method (FDM) approximates derivatives using only nearby concentration values; however, an approach that takes into account the influence of concentrations away from a point is highly intuitive and requires careful consideration. In this paper we extend the integral formulation for the case when external limitations are present. We also propose a new approach based on the use of two-dimensional Green''s function for effectiveness factor calculations in two-dimensional geometries, and provide a generic procedure for concentration profile calculation in one- and two-dimensional geometries. The proposed approach retains the representation rigor of analytical approach and reduces the computational complexities associated with FDM. It has been shown that the presence of external transport limitation shall modify the Green''s function obtained considering only internal transport limitations. The problem of very high sensitivity of the boundary contribution term in the Green''s Function Solution for two-dimensional cylindrical particle is addressed by variable transformation. It has been shown in this paper that Green''s Function method provides superior performance for larger step size when compared with the FDM approach, thus providing computationally efficient approach. [Copyright &y& Elsevier]

This paper is concerned with accurate and efficient numerical methods for solving viscous and nonviscous wave problems. The paper first introduces a new second-order PR-ADI like scheme. For an efficient simulation, the scheme is also extended to a highorder compact PRADI likemethod. Both of themhave the advantages of unconditional stability, less impact of the perturbing terms on the accuracy, and being convenient to compute the boundary values of the intermediates. Besides this, the compact scheme has high-order accuracy and costs less in computational time. Numerical results are presented to show the accuracy and efficiency of the new algorithms. [ABSTRACT FROM AUTHOR]

This paper discusses three rearrangement optimization problems where the energy functional is connected with the Dirichlet or Robin boundary value problems. First, we consider a simple model of Dirichlet type, derive a symmetry result, and prove an intermediate energy theorem. For this model, we show that if the optimal domain (or its complement) is a ball centered at the origin, then the original domain must be a ball. As for the intermediate energy theorem, we show that if α,β denote the optimal values of corresponding minimization and maximization problems, respectively, then every γ in (α,β) is achieved by solving a max-min problem. Second, we investigate a similar symmetry problem for the Dirichlet problems where the energy functional is nonlinear. Finally, we show the existence and uniqueness of rearrangement minimization problems associated with the Robin problems. In addition, we shall obtain a symmetry and a related asymptotic result. [ABSTRACT FROM AUTHOR]

Abstract: There has been a recent increase in interest in the use of distributed order differential equations (particularly in the case where the derivatives are given in the Caputo sense) to model various phenomena. Recent papers have provided insights into the numerical approximation of the solution, and some results on existence and uniqueness have been proved. In each case, the representation of the solution depends, among other parameters, on Caputo-type initial conditions. In this paper we discuss the existence and uniqueness of solutions and we propose a numerical method for their approximation in the case where the initial conditions are not known and, instead, some Caputo-type conditions are given away from the origin. [Copyright &y& Elsevier]

In this paper, we have developed a new class of 2-step rational multistep methods (RMMs) in second to fifth order of accuracy. We have presented the developments of these RMMs, as well as the local truncation error and stability analysis for each RMM that we have developed. Numerical experiments have shown that all RMMs presented in this paper are suitable to solve initial value problem of various dimensions and also stiff problems. [ABSTRACT FROM AUTHOR]

This paper investigates the analytical approximate solutions of singular fourth order four-point boundary value problems using reproducing kernel method (RKM). The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the RKM can not be used directly to solve singular fourth order four-point boundary value problems (BVPs), since there is no method of obtaining reproducing kernel (RK) satisfying four-point boundary conditions. The aim of this paper is to fill this gap. A method for obtaining RK satisfying four-point boundary conditions is proposed so that RKM can be used to solve singular fourth order four-point BVPs. Results of numerical examples demonstrate that the method is quite accurate and efficient for singular fourth order four-point BVPs. [ABSTRACT FROM AUTHOR]

In this paper, one investigates the elastic flexural-torsional buckling of linearly tapered cantilever strip beam-columns acted by axial and transversal point loads applied at the tip. For prismatic and wedge-shaped members, the governing differential equation is integrated in closed form by means of confluent hypergeometric functions. For general tapered members (0<(hmax-hmin)/hmax<1), the solution to the boundary value problem is obtained in the form of a Frobenius’ series, which is shown to converge in the interior of the domain and at the boundary if and only if 0<(hmax-hmin)/hmax<1/2. Therefore, for 1/2<=(hmax-hmin)/hmax<1 the Frobenius’ series solution cannot be used to establish the characteristic equation for the cantilever beam-columns; the problem is then solved numerically by means of a collocation procedure. Some of the analytical solutions (buckling loads) were compared with the results of shell finite-element analyses and an excellent agreement was found in all cases, thus validating the mathematical model and confirming the correctness of the analytical results. The paper closes with a discussion on the convexity of the stability domain (in the load parameter space) and the accuracy of approximations based on Dunkerley-type theorems. [ABSTRACT FROM AUTHOR]

This paper presents some new results on a spectrum in a complex plane for the second order stationary differential equation with one Bitsadze-Samarskii type nonlocal boundary condition. In this paper, we survey the characteristic function method for investigation of the spectrum of this problem. Some new results on characteristic functions are proved. Many results of this investigation are presented as graphs of characteristic functions. A definition of constant eigenvalues and the characteristic function is introduced for the Sturm-Liouville problem with general nonlocal boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper we investigate spline wavelets on the interval with homogeneous boundary conditions. Starting with a pair of families of B-splines on the unit interval, we give a general method to explicitly construct wavelets satisfying the desired homogeneous boundary conditions. On the basis of a new development of multiresolution analysis, we show that these wavelets form Riesz bases of certain Sobolev spaces. The wavelet bases investigated in this paper are suitable for numerical solutions of ordinary and partial differential equations. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we show differentiability of solutions with respect to the given boundary value data for nonlinear singularly perturbed boundary value problems and its corresponding asymptotic expansion of small parameter. This result fills the gap caused by the solvability condition in Esipova’s result so as to lay a rigorous foundation for the theory of boundary function method on which a guideline is provided as to how to apply this theory to the other forms of singularly perturbed nonlinear boundary value problems and enlarge considerably the scope of applicability and validity of the boundary function method. A third-order singularly perturbed boundary value problem arising in the theory of thin film flows is revisited to illustrate the theory of this paper. Compared to the original result, the imposed potential condition is completely removed by the boundary function method to obtain a better result. Moreover, an improper assumption on the reduced problem has been corrected. [Copyright &y& Elsevier]

Wavelet transforms or wavelet analysis is a recently developed mathematical tool for many problems. Wavelets also can be applied in numerical analysis. In this paper, we apply Haar wavelet methods to solve ordinary differential equations with initial or boundary condition known. To avoid the tedious calculations and to promote the study of wavelets to beginners, we proposed a simple way to perform the calculations for the matrix representation. The procedure applied in this paper is taking the Haar Series for the highest order of differential and integrate the series. Four numerical examples are shown which including first, second, higher order differential equations with constant and variable coefficients. The results show that the proposed way are quite reasonable when compare to exact solution. [ABSTRACT FROM AUTHOR]

To circumvent the constraint in application of the conventional differential quadrature (DQ) method that the solution domain has to be a regular region, an interpolation-based local differential quadrature (LDQ) method is proposed in this paper. Instead of using regular nodes placed on mesh lines in the DQ method (DQM), irregularly distributed nodes are employed in the LDQ method. That is, any spatial derivative at a nodal point is approximated by a linear weighted sum of the functional values of irregularly distributed nodes in the local physical domain. The feature of the new approach lies in the fact that the weighting coefficients are determined by the quadrature rule over the irregularly distributed local supporting nodes with the aid of nodal interpolation techniques developed in the paper. Because of this distinctive feature, the LDQ method can be consistently applied to linear and nonlinear problems and is really a mesh-free method without the limitation in the solution domain of the conventional DQM. The effectiveness and efficiency of the method are validated by two simple numerical examples by solving boundary-value problems of a linear and a nonlinear partial differential equation. Copyright © 2007 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with stability properties of Runge–Kutta methods for the initial value problem in nonlinear neutral delay differential equationsThe new concepts of -stability, -stability and Weak -stability are introduced, and it is shown that -algebraically stable Runge–Kutta methods with piecewise linear interpolation are -, - and Weakly -stable. Two numerical examples are given in the end of this paper which confirm our results. [Copyright &y& Elsevier]

Abstract: When solving a system of first order differential equations involving boundary conditions using collocation at k Gauss points that are the zeros of , the kth degree Legendre polynomial, to produce a continuous solution over the problem interval, it is well known that the global error at the mesh points is and at other points the global error is . In addition, it is known that the error in the derivative of the solution at non-mesh points is . This paper shows that the error in the solution at the zeros of in each subinterval is and that the error in the derivative of the solution at the zeros of is . This higher order accuracy can be used in various interpolation schemes (Hermite–Birkhoff interpolation in the case that includes derivative data) to produce global errors in the solution without any additional collocation and without any additional function evaluations. Only the evaluation of the collocation solution or its derivative at certain special points is necessary. Hence, this increased accuracy is almost free compared to other methods which give global errors but at considerable extra expense. Some numerical results illustrating the higher accuracy at these special points are given at the end of the paper along with comparisons, in the form of diagrams, of results produced from the package COLSYS and the results produced from the interpolation schemes (which use COLSYS as a starting point). [Copyright &y& Elsevier]

Abstract: We construct and analyze non-standard finite difference methods for a class of singularly perturbed differential equations. The class consists of two types of problems: (i) those having solutions with layer behavior and (ii) those having solutions with oscillatory behavior. Since no fitted mesh method can be designed for the latter type of problems, other special treatment is necessary, which is one of the aims being attained in this paper. The main idea behind the construction of our method is motivated by the modeling rules for non-standard finite difference methods, developed by Mickens. These rules allow one to incorporate the essential physical properties of the differential equations in the numerical schemes so that they provide reliable numerical results. Note that the usual ways of constructing the fitted operator methods need the fitting factor to be incorporated in the standard finite difference scheme and then it is derived by requiring that the scheme be uniformly convergent. The method that we present in this paper is fairly simple as compared to the other approaches. Several numerical examples are given to support the predicted theory. [Copyright &y& Elsevier]

Abstract: The paper deals with the initial–boundary value problem for the linear wave equation where , are given constants and , , are given functions, and the unknown function and the unknown boundary value satisfy the following nonlinear integral equation: where , , , are given constants and , are given functions. In this paper, we consider three main parts. In Part 1 we prove a theorem of global existence and uniqueness of a weak solution of problem (1.1)–(1.5). The proof is based on a Galerkin method associated with a priori estimates, weak convergence and compactness techniques. For the case of , Part 2 is devoted the study of the asymptotic behavior of the solution as . Finally, in Part 3 we obtain an asymptotic expansion of the solution of the problem (1.1)–(1.5) up to order in four small parameters , , , . [Copyright &y& Elsevier]

This paper reviews the usual open boundary conditions (OBCs) for coastal ocean models and proposes a complete set of open boundaries based on stability criteria, on mass and energy conservation arguments, and on the ability to enforce external information. This set includes a radiation condition for barotropic variables, an equation of wave propagation for baroclinic velocities, and an advection equation for tracers. Considerations on mass and energy conservation properties suggest a suitable numerical treatment of the barotropic scheme, which is different from what is commonly used. Restoring terms, when classically added in the Sommerfeld OBCs, are not consistent with external fields. It is shown that this can be avoided if proper numerical schemes are used or if OBCs are applied on differences between the model and forcing rather than on absolute variables. Finally, this paper shows that simplistic advection-type methods for temperature and salinity should not be used in sigma coordinate models because this introduces errors in the computation of the horizontal pressure gradient. [ABSTRACT FROM AUTHOR]

Abstract: This paper revisits and extends the work that we started in 1972 on maximum polynomial degree (m.p.d) (k, p) Nordsieck–Gear methods for the numerical solution of the non-stiff ordinary differential equations initial value problem: f ∈ C k+2(N), N a neighborhood of the graph the exact solution values of y and its derivatives up to order p −1. In the notation (k, p) k is the number of scaled derivatives that are retained by the method, p is the order of the differential equation and k > p ⩾1. These methods belong to the class of linear multi-value predictor–corrector methods. They are fixed order, variable step and admit of accurate self-starting algorithms. In our first paper we proved that, given exact starting values for the p +1, p +2,…, k scaled derivative starting values, a vector of p Nordsieck–Gear (k, p) corrector–predictor coefficients must be recursively computed to preserve m.p.d k +1. Computational experiment showed that the sequence of coefficient vectors was well-defined in the (k,1) case for k =2,3,4,5. This led to the two-part conjecture: (1) The sequence of coefficient vectors is well-defined for all (k, p) and (2) These coefficient vectors cycle to their fixed values at most k − p +1 steps. Our second paper dealt with the construction of a starting method which yields starting scaled derivatives of order p +1, p +2,…, k, all accurate to O(h k+2), with exactly (k − p +1)(k − p +2)/2+2 evaluations of the derivative function f. In this paper we begin by proving the first conjecture and a slightly stronger version of the second. With these results in hand we then prove convergence in the (k,1) method. It is shown that the starting algorithm’s guaranteed O(h k+2) starting error for the (k,1) method can be preserved globally for sufficiently small step-size h by augmenting two corrector–predictor evaluations with a stability-preserving, computationally efficient refinement. We have coded a (5,1) Fortran 77 algorithm, DIFFEQ, to implement, test, and confirm the underlying theory. We compare its computational efficiency with the variable order, variable step Adams–Gear method contained in the 1995 edition of the International Mathematical and Statistical Library (IMSL). [Copyright &y& Elsevier]