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This paper employs a numerical approach to investigate the impact of liquidity shocks on European options in modeling markets. To accurately capture the behavior of European options under liquidity shocks, a coupled system of differential equations is employed, consisting of a degenerate parabolic equation and a diffusion-free equation. The primary focus is on developing and analyzing implicit-explicit methods for numerically simulating European option pricing, specifically considering the presence of liquidity shocks while ensuring the positivity of the solution. The paper also includes convergence analysis and establishes the discrete comparison principle for the developed methods. Numerical experiments are conducted using both uniform and nonuniform meshes to validate the theoretical findings, demonstrating the efficiency and accuracy of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, novel energy-preserving finite difference schemes are presented and analyzed for one-dimensional and two-dimensional nonlinear wave equations with variable coefficients, respectively. Initially, the original differential equations are reformulated as corresponding Hamiltonian systems. Subsequently, the semi-discrete schemes are constructed using the sixth-order finite difference formula. The discrete energy method is employed to rigorously prove the energy conservation laws, convergence, and stability of the solutions for the spatial semi-discrete scheme. Next, the time derivatives in the semi-discrete schemes are discretized with the third-order strong stability preserving Runge-Kutta method, then the local truncation error of the finite difference scheme is provided. The accuracy and effectiveness of the constructed schemes, along with the theoretical analysis, are demonstrated through numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear beam equation with the p (x) -biharmonic operator, where the exponent p (x) is a given function. We transform the problem into a system of two differential equations and prove the existence, uniqueness and regularity of the weak solution. Then we formulate the discrete problem associated with that system using the finite element method and prove the existence, uniqueness and stability of the discrete solution. We also investigate the order of convergence and prove some error estimates. Next, we apply the Lagrange basis to obtain an algebraic system of equations. Finally, we implement the computational codes in Matlab software considering the one and two dimensional cases and we present examples to illustrate the theory. • Study of a nonlinear beam equation with the p (x) -biharmonic operator. • Proof of the existence, uniqueness and regularity of the weak solution. • Discrete formulation with the mixed finite element method. • Demonstration of the existence, uniqueness and regularity of the discrete solution. • Numerical simulations with Matlab software for the one and two dimensional cases. [ABSTRACT FROM AUTHOR]

In this paper, the regularity of the solution for a multi-term time-fractional fourth-order equation is analyzed, and the derived result shows that the solution behaves a weak singularity at initial time t = 0. By introducing an intermediate variable p = − Δ u , we obtain a system of two second-order differential equations, which is discretised by using the L1 scheme on graded meshes in time and the mixed finite element method (MFEM) in space. It is shown that the optimal pointwise-in-time error estimate is derived for the proposed scheme. Moreover, the fast L1-MFEM is constructed to improve the computational efficiency. Finally, numerical experiments are provided to further verify our theoretical convergent result and the computational efficiency of the proposed two schemes. [ABSTRACT FROM AUTHOR]

Fourth-order differential equations play an important role in many applications in science and engineering. In this paper, we present a three-field mixed finite-element formulation for fourth-order problems, with a focus on the effective treatment of the different boundary conditions that arise naturally in a variational formulation. Our formulation is based on introducing the gradient of the solution as an explicit variable, constrained using a Lagrange multiplier. The essential boundary conditions are enforced weakly, using Nitsche's method where required. As a result, the problem is rewritten as a saddle-point system, requiring analysis of the resulting finite-element discretization and the construction of optimal linear solvers. Here, we discuss the analysis of the well-posedness and accuracy of the finite-element formulation. Moreover, we develop monolithic multigrid solvers for the resulting linear systems. Two and three-dimensional numerical results are presented to demonstrate the accuracy of the discretization and efficiency of the multigrid solvers proposed. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to derive superconvergence results for nonlinear Klein-Gordon-Schrödinger equation (KGSE) with backward differential formula (BDF) finite element method (FEM). A linearized fully discretized scheme is presented to approximate the solution of the nonlinear equations. To get rid of the restriction about the ratio between h and τ , a time-discrete system is recommended to split the error into temporal error and spatial error. Based on the detailed investigation, the technique of recombination for some terms gives the chance to bound the temporal errors in H 2 -norm and the spatial errors in H 1 -norm, respectively. By virtue of the Ritz projection and the interpolation operator together, superconvergence results for spatial error of order O (h 2 + τ 2) in H 1 -norm for the original variable are deduced based on the spatial errors. Finally, numerical example is provided to support the theoretical analysis. Here, h is the subdivision parameter, and τ is the time step. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a new exact solution based on the classical shell theory (CST) for free vibration of cross-ply laminated composite cylindrical shells with elastic restraint ends is proposed. The present exact solution can be summed up in the following steps: Firstly, the displacement functions are constructed by the governing differential equations with the exact closed form solutions; Then, the artificial spring technology is introduced to simulate the general boundary conditions of the two end edges of shell; Thirdly, the equation for natural frequencies is obtained by means of the method of reverberation-ray matrix (MRRM); Lastly, the vibration results are presented by the modified golden section search (MGSS) algorithm. By comparing the present method with published papers, the accuracy of present method is verified. On the basis of that, some new exact nature frequencies and mode shapes of the cross-ply laminated composite cylindrical shells with various classical boundary conditions and elastic restraints are performed and they can be served as the benchmark data for the future. [ABSTRACT FROM AUTHOR]

This paper studies the numerical solution of a two-dimensional quenching type nonlinear reaction-diffusion problem via dimensional splitting. The variable coefficient differential equation considered possesses a nonlinear forcing term, and may lead to strong quenching singularities that have profound multiphysics and engineering applications to the energy industry. Our investigations focus on the construction and preservative properties of a semi-adaptive Peaceman-Rachford procedure for solving the aforementioned problem. In the study, the multidimensional differential equation is decomposed into single dimensional subequations so that the computational efficiency can be effectively raised. Temporal adaptation is implemented through arc-length estimations of the rate-of-change of the numerical solution. The positivity, monotonicity and localized linear stability of the variable step splitting scheme are analyzed. These ensure key preservations of underlying physical features of the computed approximations for applications. Computational experiments are presented to illustrate our results as well as to demonstrate the viability and accuracy of the splitting method for solving singular quenching-combustion problems. [ABSTRACT FROM AUTHOR]

Our aim through this paper is to describe a Krylov based projection method in order to reduce high-order dynamical systems. We focus on differential algebraic equations (DAEs) of index-2 that arise from spatial discretization of Stokes equations. An efficient algorithm based on a projection technique onto an extended block Krylov subspace that appropriately allows us to construct a reduced order system is described. Numerical results are provided to confirm the performance of the derived method compared with other known ones. [ABSTRACT FROM AUTHOR]

In this paper we present some extensions of the classical SIR model with non-symmetric spatial dependence. SIR-type models usually describe the epidemic in a population, which is split into three categories, namely the susceptibles (S), the infected (I) and the recovered (R). The proposed model yields a system of partial integro-differential equations. Two methods handling the integrals of the equations are presented. We give numerical examples which show that the discrete models preserve the basic qualitative properties of the original biological process, and these results are supported by the oretical theorems. [ABSTRACT FROM AUTHOR]

In this paper we propose a new projection method to solve both large-scale continuous-time matrix Riccati equations and differential matrix Riccati equations. The new approach projects the problem onto an extended block Krylov subspace and gets a low-dimensional equation. We use the block Golub–Kahan procedure to construct the orthonormal bases for the extended Krylov subspaces. For matrix Riccati equations, the reduced problem is then solved by means of a direct Riccati scheme such as the Schur method. When we solve differential matrix Riccati equations, the reduced problem is solved by the Backward Differentiation Formula (BDF) method and the obtained solution is used to build the low rank approximate solution of the original problem. Finally, we give some theoretical results and present numerical experiments. [ABSTRACT FROM AUTHOR]

Chemical Enhanced Oil Recovery (EOR) processes comprise a number of flooding techniques aimed at increasing the operational life of mature oilfields. Among these, polymer flooding is one of the most developed; its functionality is to increment the aqueous viscosity, avoiding the formation of viscous fingering. Reservoir simulators consider this influence as well as other physical properties (e.g., adsorption, permeability reduction). However, the polymer degradation is usually not considered even though it plays a critical role in the viscosity. In this paper this mechanism is analyzed and coupled with the previously mentioned physical phenomena in order to present a complete study of their influence in the EOR process. Moreover, since a fully second-order accuracy scheme is used along with a Total Variation Diminishing (TVD) flux-limiting function, the influence of the latter on the recovery factor is also discussed. Results showed that the negative effect of the polymer adsorption was the most relevant physical phenomenon in terms of the oil recovery. Furthermore, the analysis of the discretization of the differential equations showed that traditional, linear first-order schemes created numerical diffusion affecting negatively the macroscopic sweeping efficiency, which disappeared when TVD techniques were used. Reservoir simulators allow determining the desired designing properties for future polymers in relationship with the characteristics of the oilfield to be exploited. [ABSTRACT FROM AUTHOR]

In this paper, we apply the method of approximate particular solutions (MAPS) based on the trigonometric functions to solve the fourth order partial differential equations (PDEs) through the use of particular solutions of two second order differential equations. The derivation of the closed-form particular solutions for higher order PDEs is a challenge and could be tedious. Such task can be achieved by a simple algebraic procedure to alleviate the difficulty of the derivation of particular solutions for fourth order PDEs. Since the closed-form particular solutions for the second order PDEs with constant coefficients are known, the proposed solution procedure is simple and direct. Five numerical examples are illustrated to demonstrate the feasibility and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a second-order backward differentiation formula (BDF) alternatingdirection implicit (ADI) orthogonal spline collocation (OSC) method is presented for the fractional integro-differential equation with two different weakly singular kernels. The integral terms are approximated by the second-order fractional quadrature rule suggested by Lubich. The convergence of the BDF ADI OSC method in L 2 norm is proved. Besides, we provide the numerical experiments to demonstrate the results of theoretical analysis and show the accuracy and the effectiveness of the BDF ADI OSC method. Numerical experiments also exhibit the optimal error estimates. [ABSTRACT FROM AUTHOR]

A novel strong form numerical method, Element Differential Method (EDM), is developed to solve geometrically complex mechanics problems based on triangular or tetrahedral meshes. The discretization of the structure under investigation has been based on Lagrange isoparametric quadrilateral or hexahedral elements while applying EDM. In this paper, a new family of isoparametric triangular and tetrahedral elements with a central node is proposed for EDM. A set of shape functions with analytical expressions for their first and second order partial derivatives is constructed for these triangular and tetrahedral elements, respectively. Moreover, a new element collocation scheme is proposed to establish a system of equations directly from the governing differential equations for internal nodes and traction-equilibrium equations for nodes on edges of an element. In this collocation scheme, no variational principles or virtual energy principles are required to set up the solution scheme, while no integration is needed when forming the coefficients of the system of equations. Numerical examples including standard patch tests and more practical problems are given to demonstrate the correctness of the constructed elements and the efficiency of the proposed element collocation method. [ABSTRACT FROM AUTHOR]

This paper presents a study on space fractional anomalous convective-diffusion and chemical reaction in the magneto-hydrodynamic fluid over an unsteady stretching sheet. The fractional diffusion model is derived from decoupled continuous time random walks in a heterogeneous porous medium. A novel transformation which features time finite difference is introduced to reduce the governing equations into ordinary differential ones in each time level. Numerical solutions are established by an implicit finite difference scheme. The stability and convergence of the method are analyzed. Results show that increasing fractional derivative parameter enhances concentration near the surface while an opposite phenomenon occurs far away from the wall. There is a reduction of mass transfer rate on the sheet with an increase in the fractional derivative parameter. Moreover, the numerical solutions are compared with exact solutions and good agreement has been observed. [ABSTRACT FROM AUTHOR]

Abstract: The aim of the paper is the description of fractional-order differential equations in terms of exact mechanical models. This result will be archived, in the paper, for the case of linear multiphase fractional hereditariness involving linear combinations of power-laws in relaxation/creep functions. The mechanical model corresponding to fractional-order differential equations is the extension of a recently introduced exact mechanical representation (Di Paola and Zingales (2012) [33] and Di Paola et al. (2012) [34]) of fractional-order integrals and derivatives. Some numerical applications have been reported in the paper to assess the capabilities of the model in terms of a peculiar arrangement of linear springs and dashpots. [Copyright &y& Elsevier]

Abstract: This paper is motivated from some recent papers treating the boundary value problems for impulsive fractional differential equations. We first make a counterexample to show that the formula of solutions in cited papers are incorrect. Second, we establish a general framework to find the solutions for impulsive fractional boundary value problems, which will provide an effective way to deal with such problems. Third, some sufficient conditions for the existence of the solutions are established by applying fixed point methods. Meanwhile, data dependence is obtained by using a new generalized singular Gronwall inequality. Finally, three examples are given to illustrate the results. [Copyright &y& Elsevier]

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 investigate a time-fractional semilinear differential equation with final nonlocal condition u (x , T) − (G u) (x) = ∑ j = 1 m ϑ j u (x , t j). Under some assumptions, we establish the existence and uniqueness of a mild solution in some suitable spaces of continuous functions. The results are based on some fixed point theorems. Two examples are presented to illustrate our results. [ABSTRACT FROM AUTHOR]

The spatial–temporal patterns of the solutions to a differential equation with non-local delayed feedback that models the population dynamics of a two-stage species have been studied in this paper. The effects of the maturation time and immature mobility constant on the existence and stability of both spatially homogeneous and inhomogeneous periodic solutions are investigated by studying the Hopf bifurcation at the spatially homogeneous steady state and computing center manifold. Our results show that the bifurcated homogeneous periodic solutions are stable while inhomogeneous periodic solutions will eventually tend to homogeneous periodic solutions after transient oscillations. Furthermore, increasing of the immature mobility constant will weaken the periodic oscillation and shorten the transient oscillation time. Numerical simulations are given at last to verify our results. [ABSTRACT FROM AUTHOR]

Abstract This paper is dedicated to the study of continuous-time mean–variance optimal portfolio selection problem with non-linear wealth equations under non-extensive statistical mechanics for the time-varying stochastic differential equation model. Firstly, we allow the returns and variance of risky assets are time-varying functions, which can fit the financial data better. Secondly, we consider an investor with the non-linear wealth equation. In fact, the wealth equations are not linear in many cases. The investor has to pay some taxes, which leads to a non-linear wealth equation. Moreover, since the return of the stocks price may be affected by a large investors portfolio selection, the wealth equation is non-linear in this case. Thirdly, the non-linear wealth equation driven by Tsallis distribution is constructed under non-extensive statistical mechanics, which can capture the characteristics of fat tails and aiguilles of the risky asset's return. The viscosity solution of the HJB equation for the portfolio problem is proposed by the optimal stochastic control theory and Lagrange multiplier method. Finally, the efficient portfolio strategy and efficient frontier under non-extensive statistical mechanics are obtained. Furthermore, numerical analysis and real data study are discussed to show our results. [ABSTRACT FROM AUTHOR]

Abstract In this paper we analyse, from the numerical point of view, two dual-phase-lag models appearing in the heat conduction theory. Both models are written as linear partial differential equations of third order in time. The variational formulations, written in terms of the thermal acceleration, lead to linear variational equations, for which existence and uniqueness results, and energy decay properties, are recalled. Then, fully discrete approximations are introduced for both models using the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. Discrete stability properties are proved, and a priori error estimates are obtained, from which the linear convergence of the approximations is derived. Finally, some numerical simulations are described in one and two dimensions to demonstrate the accuracy of the approximations and the behaviour of the solutions. [ABSTRACT FROM AUTHOR]

Abstract In this paper we study a multi-layer tumor model which is expressed as a free boundary problem of a system of partial differential equations. The problem consists of two elliptic equations describing the distribution of nutrient concentration and the pressure between tumor cells, respectively, in an unbounded strip-like region in which the tumor occupies. This region has two disjoint boundaries: While the lower part is fixed, the upper part, which stands for the tumor surface, can move as the tumor grows. Under certain conditions, the problem can be proved to admit a unique equilibrium which corresponds to the flat upper boundary. We first convert the model into a parabolic differential equation in certain function space. Next we compute the spectrum of the linearized problem at the equilibrium. By applying the geometric theory of parabolic differential equations in Banach spaces, we prove that if the cell-to-cell adhesiveness coefficient γ is larger than a threshold value γ ∗ , then the unique flat equilibrium is asymptotically stable, whereas in the case 0 < γ < γ ∗ the flat equilibrium is unstable. [ABSTRACT FROM AUTHOR]

Abstract: In the present paper we establish several differential superordinations regarding the new operator defined by using the generalized Sălăgean operator and Ruscheweyh derivative , , where , and , A number of interesting consequences of some of these superordination results are discussed. Relevant connections of some of the new results obtained in this paper with those in earlier works are also provided. [Copyright &y& Elsevier]

Abstract In this paper, we develop a diffusive plant invasion model with delay under the homogeneous Neumann boundary condition. First, the existence and uniqueness of a non-negative solution, persistence property, and local asymptotic stability of the constant steady states are established. Then by analyzing the associated characteristic equation, the stability of steady states and the existence of Hopf bifurcation are demonstrated. Under special circumstance, the discontinuous Hopf bifurcation is also investigated. Furthermore, the existence and non-existence of nonconstant positive steady states of this model are studied through considering the effect of large diffusivity. Finally, in order to verify our theoretical results, some numerical simulations are also included. It shows that the numerically observed behaviors are in good agreement with the theoretically proposed results. On the basis of numerical simulations, we provide some useful comparisons for the readers between our results and the existing ones, and discuss the effects of the delay and the diffusion term on dynamic behaviors. Numerical results indicate that diffusion can make the system unstable and increasing delay may cause the plant to go extinct. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with robust path tracking using flatness principles extended to fractional linear MIMO systems. As soon as the path has been obtained by means of the fractional flatness, a robust path tracking based on CRONE control is presented. Flatness in path planning is used to determine the controls to apply without integrating any differential equations when the trajectory is fixed (in space and in time). Several developments have been made for fractional linear SISO systems using a transfer function approach. For fractional systems, especially in MIMO, developments are still to be made. Throughout this paper, flatness principles are applied using polynomial matrices for fractional linear MIMO systems. To illustrate the robustness performances, a third-generation multi-scalar CRONE controller is compared to a PID one. [Copyright &y& Elsevier]

Abstract: This paper presents a new method to construct explicit bounds for the solutions of second order linear differential equations of the type ,, , which complements the method described in recent papers of the authors. The method lies in the use of Beesack’s version of Opial’s inequality. Some results on asymptotics of the solution for the case are also included. [Copyright &y& Elsevier]

Abstract: This paper deals with the pseudo-isochronicity for a class of septic differential systems. In this paper, we transform infinity into the origin so that the properties of infinity can be investigated with the methods developed for finite critical points. By calculating the singular point quantities and period constants of the origin, the problem of infinity being a pseudo-isochronous center has been solved in this case. [Copyright &y& Elsevier]

Abstract: The discrete delayed Hopfield neural networks is an extension of the discrete Hopfield neural networks. In this paper, the convergence of discrete delayed Hopfield neural networks is mainly studied, and some results on the convergence are obtained by using Lyapunov function. Several new sufficient conditions for the delayed networks converging towards a limit cycle with period at most 2 are proved in parallel updating mode. Also, some conditions for the delayed networks converging towards a limit cycle with 2-period are investigated in parallel updating mode. All results established in this paper extend the previous results on the convergence of both the discrete Hopfield neural networks, and the discrete delayed Hopfield neural networks in parallel updating mode. [Copyright &y& Elsevier]

Abstract: This paper improves the algorithm for the construction of explicit bounds for the solutions of second order linear differential equations of the type , , described in a recent paper of the authors. The main improvements come from the introduction of new functionals devised by Trevisan and from the application of an enhanced version of the mean value theorem for integrals when some functions of the integrand are monotonic. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a general high-order delay differential equation in the following form: where is a continuous periodic function defined on with period ; is a continuous function defined on and periodic to , for all . By applying Mawhin’s continuation theorem and a new inequality established in this paper, we obtain sufficient conditions for the existence of periodic solutions for the equation. Our results are new and the methods are different from previous works. Moreover, an example to illustrate the results is given. [Copyright &y& Elsevier]

Abstract: In this paper, we prove a theorem on the regular growth of the solutions of a linear differential equation, as an application of which we obtain some uniqueness theorems of an entire function sharing a finite nonzero complex number with its th derivative and a linear differential polynomial of its th derivative under the restriction of finite lower order or of finite lower hyper-order that is not a positive integer. The results in this paper improve many known results. Some examples are provided to show that the results in this paper are the best possible. [Copyright &y& Elsevier]

Abstract: In this paper, existence criteria of positive solutions to a class of nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales are obtained. The main tool used in this paper is the well-known Guo–Krasnoselskii fixed-point theorem. [Copyright &y& Elsevier]

Abstract: Nonlocal boundary value problems at resonance for a higher order nonlinear differential equation with a -Laplacian are considered in this paper. By using a new continuation theorem, some existence results are obtained for such boundary value problems. An explicit example is also given in this paper to illustrate the main results. [Copyright &y& Elsevier]

Abstract: This paper deals with the existence of positive solutions for the fourth-order nonlinear ordinary differential equation subject to the boundary conditions: where are constants such that , and . By means of a fixed-point theorem due to Krasnaselskii, some new existence results of positive solutions for the above multi-point boundary value problem are obtained, which improve the main results of Graef et al. [J.R. Graef, C. Qian, B. Yang, A three-point boundary value problem for nonlinear fourth-order differential equations, J. Math. Anal. Appl. 287 (2003) 217–233]. An example is given to demonstrate the main results of this paper. [Copyright &y& Elsevier]

Abstract: The principle of this paper is to deal with a new existence theory for positive periodic solutions to a kind of nonautonomous functional differential equations with impulse actions at fixed moments. Easily verifiable sufficient criteria are established. The approach is based on the fixed-point theorem in cones. The paper extends some previous results and obtains some new results. [Copyright &y& Elsevier]

Abstract: n this paper, the M/G/1 processor-sharing queue with disasters is given a detailed analysis by means of extending the supplementary variable method. The transient and steady-state distributions of the queue length are expressed as a simple and computable form, the Laplace-Stieltjes transform of the sojourn time is derived, and the Laplace transform of the busy period and its mean are obtained. Also, the approach developed in this paper is shown to be able to study more complicated M/G/1 processor-sharing models. [Copyright &y& Elsevier]

Abstract: This paper presents a new existence theory for single and multiple positive periodic solutions to a kind of nonautonomous Volterra integro-differential equations with impulse effects. Existence is established by using Krasnoselskii fixed-point theorem in cones. This paper extends some previous results and reports some new results about impulsive differential equation [Copyright &y& Elsevier]

Abstract: This paper introduces radial basis functions (RBF) into the collocation methods and the combined methods for elliptic boundary value problems. First, the Ritz-Galerkin method (RGM) is chosen using the RBF, and the integration approximation leads to the collocation method of RBF for Poisson''s equation. Next, the combinations of RBF with finite-element method (FEM), finite-difference method (FDM), etc., can be easily formulated by following Li [1] and Hu and Li [2,3], but more analysis of inverse estimates is explored in this paper. Since the RBFs have the exponential convergence rates, and since the collocation nodes may be scattered in rather arbitrary fashions in various applications, the RBF may be competitive to orthogonal polynomials for smooth solutions. Moreover, for singular solutions, we may use some singular functions and RBFs together. Numerical examples for smooth and singularity problems are provided to display effectiveness of the methods proposed and to support the analysis made. [Copyright &y& Elsevier]

Abstract: When semiexplicit differential-algebraic equations are solved with implicit Runge-Kutta methods (RK), the computational effort is dominated by the cost of solving the nonlinear systems, and therefore it is important to have good starting values to begin the iterations. For semiexplicit index-2 DAEs, starting algorithms without additional cost for RK methods with regular matrix coefficient were studied in a previous paper. However, the regularity condition on the matrix coefficient excludes some interesting methods like Lobatto IIIa and ESDIRK methods. In this paper, we study starting algorithms, without additional computational cost, for a class of Runge-Kutta methods in the case of index-2 DAEs. [Copyright &y& Elsevier]

Abstract: In this paper, the existence of nonoscillatory solutions of the first-order neutral delay differential equations with variable coefficients and delays are studied. Some new sufficient conditions are given. In particular, conditions given in this paper are weaker than those known, so the results in this paper have wider application than the existing ones. [Copyright &y& Elsevier]

Abstract: Singularly perturbed boundary value problems often have solutions with very thinlayers in which the solution changes rapidly. This paper concentrates on the case where theses layers occur near the boundary, although our method can be applied to problems with interior layers. One technique to deal with the increased resolution requirements in these layers is the use of domain transformations. A coordinate stretching based transform allows to move collocation points into the layer, a requirement to resolve the layer accurately. Previously, such transformations have been studied in the context of finite-difference and spectral collocation methods. In this paper, we use radial basis functions (RBFs) to solve the boundary value problem. Specifically, we present a collocation method based on multiquadric (MQ) functions with an integral formulation combined with a coordinate transformation. We find that our scheme is ultimately more accurate than a recently proposed adaptive MQ scheme. The RBF scheme is also amenable to adaptivity. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the monotone method on the periodic boundary value problems for integrodifferential equations of Volterra type. This problem has been studied by several authors[1–6]. The purpose of this paper is to relax the restrictions as much as possible on the coefficients M and N that appear in the one-sided Lipschitz condition. [Copyright &y& Elsevier]

Abstract: The main object of this paper is to investigate several geometric properties of the solutions of the following second-order linear differential equation: where the functions a(z) and b(z) are analytic in the open unit disk U. Relevant connections of the results presented in this paper with those given earlier by, for example, Robertson, Miller and Saitoh are also considered. [Copyright &y& Elsevier]

This paper deals with a new existence theory for positive periodic solutions to a kind of nonautonomous functional differential equation by employing the fixed-point theorem in cones. Applying the general theorems established to several biomathematical models, the paper improves some previous results and obtains some new results. [Copyright &y& Elsevier]

This paper deals with the second-order nonlinear difference equation where {ifrk} and {qk} are positive real sequences defined on NU {0}, and the nonlinearity g: R → R is nonnegative and nontrivial. Sufficient and necessary conditions are given, for the existence of bounded solutions starting from a fixed initial condition uo. The same dynamic, with f instead of g such that uf(u) > 0 for u ≠ 0, was recently extensively investigated. On the contrary, our nonlinearity g is of a small appearance in the discrete case. Its introduction is motivated by the analysis of wavefront profiles in biological and chemical models. The paper emphasizes the many different dynamical behaviors caused by such a g with respect to the equation involving function f. Some applications in the study of wavefronts complete this work. [Copyright &y& Elsevier]

This paper is concerned with the construction and study of specific finite and infinite integral transforms arising from a singular Sturm-Liouville problem on a half line, which is connected to the hydrogen atom equation. The paper goes further to describe fully the image of some spaces of square integrable functions with respect to some measure under the finite and infinite integral transforms. [Copyright &y& Elsevier]

In this paper, scheduling of divisible loads in a bus network is considered. The objective is to minimize the processing time by including the overhead component due to start-up time that could degrade the performance of the system, in addition to the inherent communication and computation delays. These overheads are considered to be constant additive factors to the communication and computation components. A closed-form expression for optimal processing time is derived. Using this closed-form expression, this paper analytically proves significant results regarding the optimal sequence of load distribution and optimal number of processors. Numerical examples are presented to illustrate the analysis. [Copyright &y& Elsevier]

Singularly perturbed two-point boundary value problems (SPBVPs) for third-order ordinary differential equations (ODEs) with a small parameter multiplying the highest derivative are considered. A numerical method is suggested in this paper to solve such problems. In this method, the given BVP is transformed into a weakly coupled system of two ODEs subject to suitable initial and boundary conditions. Then, the computational method, presented in this paper, is applied to this system. In this method, we reduce the weakly coupled system into a decoupled system. Then, to solve this decoupled system numerically, we apply a ‘boundary value technique (BVT)’, in which the domain of definition of the differential equation is divided into two nonoverlapping subintervals called inner and outer regions. Then, we solve the decoupled system over these regions as two point boundary value problems. An exponentially fitted finite difference scheme is used in the inner region and a classical finite difference scheme, in the outer region. The boundary conditions at the transition point are obtained using the zero-order asymptotic expansion approximation of the solution of the problem. This computational method is distinguished by the facts that the decoupling reduces the computational time very much and it is well suited for parallel computing. This method can be extended to a system of two ordinary differential equations, of which, one is of first order and the other is of second order. Numerical examples are given to illustrate the method. [ABSTRACT FROM AUTHOR]