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In the present paper, the initial value problem for the third order partial differential equation with time delay is studied. The first and second order of accuracy difference schemes for the numerical solution of the third order partial differential equation with time delay are presented. The illustrative numerical results are provided. [ABSTRACT FROM AUTHOR]

The Fourier series, Laplace and Fourier transforms are applicable for the solution of parabolic type involutory differential problem with constant or polynomial coefficients. In the present paper, the first and second order of accuracy difference schemes for the numerical solution of the initial boundary value problem for one dimensional parabolic type involutory partial differential equation are presented. Numerical results are given. [ABSTRACT FROM AUTHOR]

This paper reports a new numerical method based on Bernoulli polynomials for the solution of Helmholtz equations. The method uses matrix forms of Bernoulli polynomials and their derivatives by means of collocation points. Aim of this paper is to solve Helmholtz equations using this matrix relations. [ABSTRACT FROM AUTHOR]

This paper presents a study on the dissemination of information from the source node by establishing information potentials that uses the information gradient descent method. The information gradient is computed by solving the discrete approximation of a partial difference equation. The solution to this equation is the harmonic functions which are also known as harmonic information potentials. Commonly used iterative method for solving partial difference equation to obtain the harmonic information potentials was the classical Jacobi that was found to be too slow when it dealt with large domain. This study proposes faster iterative methods using Gauss-Seidel and Successive Overrelaxation with Red-Black ordering strategy (GS-RB and SOR-RB), as well as Modified Successive Overrelaxation (MSOR) schemes. The experimental results show that the execution time of GS-RB, SOR-RB and MSOR are faster than the existing methods. The advantage of the proposed methods in terms of computational speed is clearly shown with increasingly large domain size. [ABSTRACT FROM AUTHOR]

We present an enhancement of the multiscale finite element method (MsFEM) by combining it with the hp-adaptive FEM. Such a discretization-based homogenization technique is a versatile tool for modeling heterogeneous materials with fast oscillating elasticity coefficients. No assumption on periodicity of the domain is required. In order to avoid direct, so-called overkill mesh computations, a coarse mesh with effective stiffness matrices is used and special shape functions are constructed to account for the local heterogeneities at the micro resolution. The automatic adaptivity (hp-type at the macro resolution and h-type at the micro resolution) increases efficiency of computation. In this paper details of the modified MsFEM are presented and a numerical test performed on a Fichera corner domain is presented in order to validate the proposed approach. [ABSTRACT FROM AUTHOR]

A family of symmetric three-stages eight-step embedded (S3SESME) methods with eliminated phase-lag and its derivatives are described in this paper. Since the methods have three stages are belonged to the category of hybrid methods. For the new proposed family of methods we request elimination of the phase-lag and its derivatives. We note here that for this specific family of embedded pairs the phase-lag and the stability analysis are studied on the whole scheme and no on each stage of each scheme. The obtained families of embedded pairs are applied on the approximate solution of the Schrödinger equation and other related problems. The theoretical and numerical achievements show the effectiveness of the new developed embedded pairs. [ABSTRACT FROM AUTHOR]

Embedded predictor-corrector methods with two-stages of prediction and with vanished phase-lag and its derivatives are described in this paper. We give the symbol (EP²CM) since these methods have two-stages of prediction. The first stage of the predictor of the new algorithm is based on the linear eight-step symmetric method of Quinlan-Tremaine [1]. The new scheme is non-linear since has three-stages. These methods can be used on the approximate solution of: 1. initial-value problems (IVPs) with oscillatory solutions, 2. boundary-value problems (IVPs) with oscillatory solutions, 3. orbital problems 4. the Schrödinger equation and related problems. The new presented algorithm belongs to the embedded methods. The numerical and theoretical achievements show the efficiency of the new produced embedded scheme. [ABSTRACT FROM AUTHOR]

Modelling and simulation of thermal processes, based on the principles of classical thermodynamics, requires numerical analysis of partial differential equations of evolution of the parabolic type. This paper demonstrates how the generalized Fourier method can be applied to the development of robust and effective computational algorithms, with the direct application to the design and performance of buildings with controlled energy consumption. [ABSTRACT FROM AUTHOR]

The paper considers the problem of estimating the stress-strain state of a cylindrical pipe section according to measured vertical displacements of a number of cross-sections of the cylinder. The problem is ambiguous, as there are an infinite number of different variants of loading, under the action of which we obtain the same values of displacements at a finite number of the set points of the pipeline. Four variants of such loading from the hard(concentrated load) to soft (continuously distributed according to some rule) are discussed . As a model of a cylindrical pipe section is used the linear theory of thin shells. The method of orthogonal sweep Godunov is used for the numerical solution of the corresponding linear boundary-value problems. [ABSTRACT FROM AUTHOR]

This paper concerns the numerical solution of the Black-Scholes PDE with a Neumann boundary condition on the right boundary. We consider finite difference schemes for the semi-discretization, which leads to a system of ODEs with corresponding matrix M. In this paper stability bounds for exp(tM) (t ≥ 0) are proved. A scaled version of the Euclidean norm, denoted by | · |H is considered. The advection and diffusion term of the PDE are analyzed separately. It turns out that the Neumann boundary condition leads to a growth of |exp(tM)|H with the number of grid points m for the pure advection problem. [ABSTRACT FROM AUTHOR]

The generalised Burgers' equation has been subject to a considerable amount of research on how the equation should behave according to asymptotic analysis, however there has been limited research verifying the asymptotic analysis. In order to verify the asymptotic analysis, this paper aims to run long time and detailed numerical simulations of Burgers' equation by employing suitable rescalings of Burgers' equation. It is hoped that this technique will make it possible to notice subtle changes in the shock structure which would otherwise be impossible to observe. The main aim of this paper is to validate the numerical methods used in order to allow further research into shock evolution where further relaxation effects will be included. [ABSTRACT FROM AUTHOR]

In this paper, some elements of the theory of the Wright function φ are discussed. The Wright function-along with the Mittag-Leffler function-plays a prominent role in the theory of the partial differential equations of the fractional order that are actively used nowadays for modeling of many phenomena including e.g. the anomalous diffusion processes or in the theory of the complex systems. This function appears there simultaneously as a Green function in the initial-value problems for the model linear equations with the constant coefficients and as a special solution invariant under the groups of the scaling transformations of the fractional differential equations. In this paper, both of these applications are shortly introduced. Whereas the analytical theory of the Wright function is already more or less well developed, its numerical evaluation is still an area of the active research. In this paper, the numerical evaluation of the Wright function is discussed with a focus on the case of the real axis that is very important for applications. In particular, several approaches are presented including the method of series summation, integral representations, and asymptotical expansions. In different parts of the complex plane different numerical techniques are employed. In each case, estimates for accuracy of the computations are provided. [ABSTRACT FROM AUTHOR]

This paper presents a new fast multipole boundary element method (FMBEM) for multiple scattering analyses in time-domain. The conventional time-domain BEM has two disadvantages: 1) The numerical instability in time stepping procedure, and 2) much computational time and memory. Our proposed method overcomes these two disadvantages using the convolution quadrature method for 1) and the fast multipole method for 2). As numerical examples, the large-scale multiple scattering of SH waves by many inclusions is conducted by the proposed method in this paper. [ABSTRACT FROM AUTHOR]

This paper presents numerical simulation results of the deformation and breakup of an isolated liquid drop suspended in an ambient, immiscible viscous fluid under shear flow. The model predicting the dynamic behavior of the drop is based on a diffuse interface method. The interface between the two fluids is tracked by an order parameter, namely the mass concentration. The fully transient, three dimensional Navier-Stokes equations for an incompressible fluid are solved by a projection method on a fixed Cartesian grid which the interface moves through to ensure accurate calculation of the surface evolution. Surface tension effects are incorporated into the model through a modified stress. This paper focuses on steady shape analysis and the end pinching mechanism of drop breakup. The numerical results of drop deformation and breakup show very good agreement with theoretical analysis and experimental observations, which indicate that the diffuse interface method can successfully capture the main behavior of the drop deformation and breakup Detailed instructions for typing your article are given in the following. [ABSTRACT FROM AUTHOR]

This paper poses a magnetic field problem in cylindrical coordinate for two regions with different permeability for each one. The linear partial differential equation governing this problem is in the form of Laplace and Poisson equation. This problem is then solved using classical Gauss-Seidel Algorithm for the Finite difference (FD) solution of linear partial differential equation. In order to obtain adequate solution for a reasonable number of grid points for the regions under consideration a considerable amount of time will take for the program to converge. The paper presents a different technique known as Weighted Multi-Grid which will speed up the convergence process. In this method, the solution to the differential equation between two grid points for obtaining the initial condition is considered to be linear in nature with different weight for the value of each grid point. This problem is then solved for the minimum number of grid points let’s say one point in each region plus the boundary points. The initial guess for each new point for every level of computation is found as the weighted average of the two adjacent points. It then continues with finding the optimum weighting values for Laplace’s or Poisson’s equation. The main contribution is made by regarding the effect of the optimum initial weighted values of the variable vector in the convergence time for the Gauss-Seidel algorithm. [ABSTRACT FROM AUTHOR]

In recent years, intensive research has been done at the Brno University of Technology Faculty of Information Technology Department of Intelligent Systems in the field of numerical solutions of systems of ordinary and partial differential equations. The basic numerical method employed is the so-called Modern Taylor Series Method (MTSM). It has been described, studied, and numerous aspects have been investigated such as processing in parallel systems. Also a simulation system TKSL has been developed which is based on the Taylor series method. For some results see [1], [2]. Although there have been considerable practical results, theoretical issues are yet to be investigated since the underlying method has been devised by analogy with analogue solvers of such systems. In this paper we provide only a basic idea of a theoretical background. The MTSM is based on a transformation of the initial problem into another initial problem with polynomials on the right-hand sides. This is a precondition for a Taylor series method to be successfully applied to the task of finding a numerical solution. The solution of the transformed initial problem then includes the solution of the original original system. A transformation similar to that described in this paper can be found in [3]. Practical aspects are discussed in [4], and [1]. An outline of the special type transformations for functions commonly encountered in initial problems can be found in [5]. The MTSM uses various types of recurrent formulae that can be used to calculate the Taylor series terms of the unknown functions. They include the detection of polynomial solutions.Another type of recurrent formulae clears the way for parallel programs to be used for solving. This type might also be of some interest for further research since it defines another type of initial problem where the right-hand sides include only powers of unknown functions. [ABSTRACT FROM AUTHOR]

In this paper, the methods of homotopy perturbation sumudu transform is mentioned. Linear and nonlinear Schrödinger equations is solved HPSTM and obtained exact solutions. [ABSTRACT FROM AUTHOR]

The purpose of the talk is the presentation of some recent advances in the numerical solution of differential equations, with special emphasis to reaction-diffusion problems, Hamiltonian problems and ordinary differential equations with discontinuous right-hand side. As a special case, in this short paper we focus on the solution of reaction-diffusion problems by means of special purpose numerical methods particularly adapted to the problem: indeed, following a problem oriented approach, we propose a modified method of lines based on the employ of finite differences shaped on the qualitative behavior of the solutions. Constructive issues and a brief analysis are presented, together with some numerical experiments showing the effectiveness of the approach and a comparison with existing solvers. [ABSTRACT FROM AUTHOR]

Consider the following problem related to an ordinary differential equation: 'For given constans ua,ub,M,N, function f(x) in [a,b], find ∂u/∂n at x = a,b which satisfies u"+Mu′+Nu = f in (a,b),u(a) = ua,u(b) = ub'. This kind of problem is called the "Dirichlet-Neumann map problem". This problem is usually solved in two steps. The solution is obtained in the first steps. In the second step, u′(a), u′(b) is computed by differentiating the solution. However, this two-step procedure is inefficient because the solution in (a,b) obtained in the first step is not essentially required. In this paper, the author presents a new strategy for obtaining Neumann data directly via a boundary integral equation formulation. Using this strategy, an explicit analytical expression of the Dirichlet-Neumann map of this problem can be directly obtained by solving 2 × 2 matrices. Furthermore, an extension of the strategy to partial differential equations in two-dimensional space and numerical algorithms is also presented. [ABSTRACT FROM AUTHOR]

One of the shortcomings of the Black and Scholes model on option pricing is the assumption that trading the underlying asset does not affect the underlying asset price. This can happen in perfectly liquid markets and it is evidently not viable in markets with imperfect liquidity (illiquid markets). It is well-known that markets with imperfect liquidity are more realistic. Thus, the presence of price impact while studying options is very important. This paper investigates a solution for the option pricing PDE in illiquid markets using the homotopy analysis method. [ABSTRACT FROM AUTHOR]

In this paper, we propose the meshless approach for the numerical solution of time dependent partial differential algebraic equations (PDAEs) in terms of finite difference scheme generated from radial basis functions (RBF-FD). Using the method, we can circumvent the influence from an index jump of PDAEs in some degree. Numerical example shows that the method has some advantages over some known methods, such as less computation and more accurate for PDAEs. [ABSTRACT FROM AUTHOR]

This paper deals with stability in the numerical solution of general one-dimensional partial differential equations with variable coefficients. We will generalize stability results for central finite difference schemes on non-uniform grids that were obtained by In’t Hout & Volders (2009) for the Black-Scholes equation. Subsequently we will apply our stability results to the CEV model. [ABSTRACT FROM AUTHOR]

This paper discusses a discretisation scheme which is based on point collocation and integrated radial basis function networks (IRBFNs) for the solution of elliptic differential equations (DEs). The use of IRBFNs to represent the field variable in a given DE gives several advantages over the case of using conventional RBFNs and polynomials. Some numerical examples are included for demonstration purposes. [ABSTRACT FROM AUTHOR]

A comparison study of algorithm on a rectangular collocated mesh is conducted for variable density Navier-Stokes equations with continuous surface forces. The algorithms include the original projection method (AI-TI) with the surface force calculated only in the predictor steps, the named balanced-force projection method (AII-TIV) with the surface force and the pressure gradient calculated together, and a consistent projection method (AIII-TVII) developed in this paper. Detailed comparisons are also conducted among the techniques for calculation of the pressure gradient and surface force at a cell center. A consistent projection method updates the velocity at a cell center in a very difference way with the balanced-force projection formula. A conservative interpolation is used to update the velocity a cell center, which is further used to obtain the sum of the pressure gradient and the surface force. [ABSTRACT FROM AUTHOR]

The dynamic response of an elastic beam attached to a spring rigid bar of two degrees of freedom is studied in this paper. The partial differential equation (PDE) describing the beam and the ordinary differential equations (ODEs) concerning the rigid bar are transformed into a homogeneous first order system of equations. The eigenfrequencies and mode shapes of the beam are determined and the problem is solved by the matrix exponential method. A numerical experiment based on true-life values of the parameters is presented. The results are verified by Finite Element (FE) code modeling. An optimization problem with respect to the system parameters is solved. [ABSTRACT FROM AUTHOR]

In this paper we will present an effective numerical method for the Black-Scholes equation with transaction costs for the limiting price u(s, t;a). The technique combines the Rothe method with a two-grid (coarse-fine) algorithm for computation of numerical solutions to initial boundary-values problems to this equation. Numerical experiments for comparison the accuracy ant the computational cost of the method with other known numerical schemes are discussed. [ABSTRACT FROM AUTHOR]

This paper deals with numerical methods for high frequency wave scattering. It introduces a new hybrid technique that couples a directional fast multipole method for a subsection of a scattering surface to an asymptotic formulation over the rest of the scattering domain. The directional fast multipole method is new and highly efficient for the solution of the boundary integral formulation of a general scattering problem but it requires at least a few unknowns per wavelength on the boundary. The asymptotic method that was introduced by Bruno and collaborators requires much fewer unknowns. On the other hand the scattered field must have a simple structure. Hybridization of these two methods retains their best properties for the solution of the full problem. Numerical examples are given for the solution of the Helmholtz equation in two space dimensions. [ABSTRACT FROM AUTHOR]

This article describes a parallel computational technique to accelerate radial point interpolation method (RPIM)-based meshfree method using graphics hardware. RPIM is one of the meshfree partial differential equation solvers that do not require the mesh structure of the analysis targets. In this paper, a technique for accelerating RPIM using graphics hardware is presented. In the method, the computation process is divided into small processes suitable for processing on the parallel architecture of the graphics hardware in a single instruction multiple data manner. [ABSTRACT FROM AUTHOR]

The dynamic response of a flexible beam mounted to an elastically supported mass is the subject of study in the paper. The Finite Element Method is used for transforming the partial differential equation describing the transverse vibrations of the beam into an ODE system to which the equations concerning the mass-spring system are added. The matrix exponent method is applied for the solution of the equations. A numerical example is proposed. [ABSTRACT FROM AUTHOR]

This paper deals with ADI type schemes in the numerical solution of the large systems of stiff ODEs that arise after spatial discretization of the Heston PDE from financial option pricing. A feature of this well-known two-dimensional PDE is the presence of a mixed spatial derivative term, and ADI schemes were not originally developed to handle such terms. We discuss how to adapt three ADI schemes and next we review theoretical results and perform numerical experiments. [ABSTRACT FROM AUTHOR]

The electrical activity of the heart is often modelled by a coupled system of partial differential equations and ordinary differential equations called the bidomain equations. Obtaining an accurate numerical solution to this model is extremely computationally expensive. In this paper we present a novel approach that is computationally competitive and combines the use of spectral elements on simplices for the spatial discretisation and a third order linearly implicit time integration scheme. All known approaches to solve this problem have concentrated on the use of low order methods. Our spatial discretisation is a high order continuous Galerkin method which is coupled to an adaptive linearly implicit Rosenbrock time integration scheme. The scheme presented allows the time steps used to be chosen by accuracy considerations rather than stability considerations and allows for highly accurate numerical simulations. [ABSTRACT FROM AUTHOR]

Meshless analysis based on the radial point interpolation method (RPIM) is effective for solving partial differential equations without a mesh structure of the analysis target. An alternative method that yields fast computation, called the modified radial point interpolation method (MRPIM), has been proposed for meshless analysis. Both methods are based on the idea that basis functions of the solution, called shape functions, are represented using radial basis functions (RBFs). The main difference between them is in the definition of the shape functions. In particular, MRPIM defines the shape functions at each small subdomain generated as a uniform division of the target domain, whereas RPIM does not require such divisions. This division does not work effectively for analysis with non-uniformly distributed nodes, since the subdomain construction process is independent of the node density. This paper presents a new RBF-based meshless method that is applicable to non-uniformly distributed nodes. [ABSTRACT FROM AUTHOR]

We construct a discrete fundamental solution for the parabolic Dirac operator which factorizes the non-stationary Schrödinger operator. With such fundamental solution we construct a discrete counterpart for the Teodorescu and Cauchy-Bitsadze operators and the Bergman projectors. We finalize this paper with convergence results regarding the operators and a concrete numerical example. [ABSTRACT FROM AUTHOR]

This paper numerically investigated the two-dimensional motion of a bead in shallow water down a steep rough bed via smoothed particle hydrodynamics (SPH). The motivation of this work is to study the mechanism of particle transport in shallow flows. The simulated results show agreement with the experimental data which also verifies the validity of SPH. Moreover, a series of numerical experiments are carried out to discuss the influences of bead material, slope gradient on the characteristics of bead motion and the state of shallow flow. [ABSTRACT FROM AUTHOR]