Problem Of First Kind Research Articles

On the approximate solutions of a class of fractional order nonlinear Volterra integro-differential initial value problems and boundary value problems of first kind and their convergence analysis

In this work we consider a class of fractional order Volterra integro-differential equations of first kind where the fractional derivative is considered in the Caputo sense. Here, we consider the initial value problem and the boundary value problem separately. For simplicity of the analysis, we reduce each of these problems to the fractional order Volterra integro-differential equation of second kind by using the Leibniz’s rule. We have obtained sufficient conditions for the existence and uniqueness of the solutions of initial and the boundary value problems. An operator based method has been considered to approximate their solutions. In addition, we provide a convergence analysis of the adopted approach. Several numerical experiments are presented to support the theoretical results.

High accuracy two-level implicit compact difference scheme for 1D unsteady biharmonic problem of first kind: application to the generalized Kuramoto–Sivashinsky equation

ABSTRACTWe propose a two-level implicit high order compact scheme for the one-dimensional unsteady biharmonic problem of first kind. The values of φ and are prescribed on the boundary. Using a combination of values of φ and at each grid point, the difference formula is derived for the unsteady biharmonic equation without discretizing the boundary conditions. The proposed method has second order time accuracy and fourth order space accuracy using just three grid points of a single compact stencil at every time level. The first order space derivative is also computed with same accuracy as a by-product of the method. Using the Von-Neumann analysis, the derived scheme is shown to be unconditionally stable. With a slight modification, the proposed method is applicable to solve singular problems. The performance of the proposed scheme is illustrated by numerical experiments done on a collection of test problems having physical significance including the nonlinear Kuramoto–Sivashinsky equation.

Two-level implicit high order method based on half-step discretization for 1D unsteady biharmonic problems of first kind

In this work, compact difference scheme based on half-step discretization is proposed to solve fourth order time dependent partial differential equations subject to Dirichlet and Neumann boundary conditions. The difference method reported here is second order accurate in time and fourth order accurate in space. The scheme employs only three grid points at each time-level and the given boundary conditions are exactly satisfied with no further approximations at the boundaries. The linear stability of the presented method has been examined and it is shown that the proposed two-level finite difference method is unconditionally stable for a model linear problem. The developed method is directly applicable to fourth order parabolic partial differential equations with singular coefficients which is the main highlight of our work. The method is successfully tested on singular problem. The proposed method is applied to find the numerical solution of the classical nonlinear Kuramoto–Sivashinsky equation and the extended Fisher–Kolmogorov equation. Comparison of the obtained results with those of some earlier known methods demonstrate the superiority of the present approach.

Możliwości i ograniczenia ilościowej analizy zawartości baz danych wizualnych. Schemat analizy fotografii zebranych w ramach projektu „Niewidzialne miasto”

The article presents possibilities and limitations of quantitative analysis of databases of visual data (in thiscase it is analysis of database of photos). The axis of the article is analysis of database of “Invisible city”project (web page: http://www.niewidzialnemiasto.pl/), its first goal was photographic documentation ofgrassroots forms of modification of city space done by inhabitants of large Polish cities (broadly speakingthis informal activity reduces to beautification of life environment and to improvement of its functionality).The moment the database of “Invisible city” project had grown to the size of several thousands of pictures, the idea arose that collected material can be analyzed in a quantitative way, which would enable more general, systematic view on phenomenon interesting for researchers.In the article, firstly, there are presented tools, scheme and general results of analysis of photos gathered indatabase of “Invisible city” project, here, in other words, is presented the scheme of transformation ofvisual data to numerical data. Secondly, there are identified moments where the scheme of quantitativeanalysis of “Invisible city” fails, where it may result in information loss, doesn’t give us one hundredpercent certainty for obtained results (problems of first kind are connected with dynamics of the formationand growth of database, problems of second kind arise at the stage of pictures analysis). Thirdly, there isproposed and described one potential master (meta)analysis, that we can impose on analysis of databaseof visual data of “Invisible city” project in order to check its accuracy. Despite of mentioned above problems,difficulties associated with the fact that database of “Invisible city” was created spontaneously to some extent (it was balancing between database and photo gallery sometimes), results of quantitative analyzes are reliable, they reconstruct the dynamics of invisible city with accuracy and give strong foundation for in-depth description.

A Two Level Implicit Difference Formula of O(k2+h4) for the Numerical Solution of One Space Dimensional Unsteady Quasi-Linear Biharmonic Problem of First Kind

A Two Level Implicit Difference Formula of O(k2+h4) for the Numerical Solution of One Space Dimensional Unsteady Quasi-Linear Biharmonic Problem of First Kind

High Accuracy Difference Formulae For A Fourth Order Quasi-Linear Parabolic Initial Boundary Value Problem Of First Kind

In this paper, new three level implicit finite difference methods of O(k^2+h^2) and O(k^2+h^4) are proposed for the numerical solution of fourth order quasi-linear parabolic partial differential equations in one space variable, where k\\gt 0 and h\\gt 0 are grid sizes in time and space coordinates respectively. In both cases, we use only nine grid points. The numerical solution of \\partial u/\\partial x is obtained as a by-product of the method. The characteristic equation for a model problem is established. Application to a linear singular equation is also discussed in detail. Four examples illustrate the utility of the new difference methods.

Block iterative methods for the numerical solution of three dimensional mildly non-linear biharmonic problems of first kind

In this article, we discuss two sets of new finite difference methods of order two and four using 19 and 27 grid points, respectively over a cubic domain for solving the three dimensional nonlinear elliptic biharmonic problems of first kind. For both the cases we use block iterative methods and a single computational cell. The numerical solution of (∂u/∂n) are obtained as by-product of the methods and we do not require fictitious points in order to approximate the boundary conditions. The resulting matrix system is solved by the block iterative method using a tri-diagonal solver. In numerical experiments the proposed methods are compared with the exact solutions both in singular and non-singular cases.

A fourth-order finite difference method for the general one-dimensional nonlinear biharmonic problems of first kind

We present two new finite difference methods of order two and four in a coupled manner for the general one-dimensional nonlinear biharmonic equation y IV= f( x, y, y′, y″, y″′) subject to the boundary conditions y(a)=A 0, y′(a)=A 1, y(b)=B 0,y′(b)=B 1 . In both cases, we use only three grid points and do not require to discretize the boundary conditions. First-order derivative of the solution is obtained as a by-product of the methods. The methods are successfully applied to the problems both in cartesian and polar coordinates. Numerical examples are given to illustrate the methods and their convergence.

Finite difference methods of order two and four for 2-d non-linear biharmonic problems of first kind

For the numerical integration of the 2-D nonlinear biharmonic problems of first kind, we report two difference methods of order two and four over a rectangular domain. These methods use only the nine grid points and do not require fictitious points in order to approximate the boundary conditions. Derivatives of the solution are obtained as a by-product of the methods. In numerical experiments, the new second and fourth order formulas are compared with the exact solutions.