Finite difference surface representation considering effect of boundary curvature

Partial differential equations of the second- and higher-order are encountered in a number of areas of importance to engineering and mathematical physics. As the processes that are accounted for in the development of the governing differential equations increase in complexity, the differential equations themselves tend to acquire a higher-order. The theory of second-order partial differential equations has found extensive applications in the study of problems in fluid mechanics, flow in porous media, heat conduction in solids, diffusive transport of chemicals in porous media, wave propagation in strings and membranes, and in mechanics of solids. Higher-order partial differential equations have been developed particularly in relation to studies in mechanics of solids. These include the formulation of the biharmonic equation for the solution of problems in plane elasticity and in the flow of viscous fluids, the fourth-order partial differential equations associated with the flexural mechanics of prismatic plates, the sixth-order partial differential equations used to describe the mechanics of shell structures and other fourth-order partial differential equations which describe the mechanics of plastic solids. In this chapter we shall restrict attention to the examination of second-order partial differential equations.

On the variational iteration method and other iterative techniques for nonlinear differential equations

The purpose of this study is to develop a numerical method for bifurcation analysis of nonlinear partial differential equations, based on the reduction of partial differential equations to ordinary ones, using the Kolmogorov-Arnold theorem. Methods. This paper describes a method for reducing partial differential equations to ordinary ones using the Kolmogorov-Arnold theorem, as well as methods for the bifurcation analysis of nonlinear boundary value problems for ordinary differential equations. Results. The paper presents a new method for solving and bifurcation analysis of nonlinear boundary value problems for partial differential equations, which allow variational formulation. The method was applied to a nonlinear two-dimensional Bratu problem with Dirichlettype boundary conditions. Conclusion. A new method of bifurcation analysis for nonlinear partial differential equations has been developed. Specifically, a method has been proposed for reducing partial different equations to ordinary equations, which allows the use of the developed apparatus of bifurcation analysis for boundary value problems of ordinary differential equations. This method allows conducting bifurcation analysis for arbitrary nonlinear partial differential equations.

This work presents a review and analysis of modern asymptotic methods for analysis of singularly perturbed problems with interior and boundary layers. The central part of the work is a review of the work of the author and his colleagues and disciples. It highlights boundary and initial-boundary value problems for nonlinear elliptic and parabolic partial differential equations, as well as periodic parabolic problems, which are widely used in applications and are called reaction–diffusion and reaction–diffusion–advection equations. These problems can be interpreted as models in chemical kinetics, synergetics, astrophysics, biology, and other fields. The solutions of these problems often have both narrow boundary regions of rapid change and inner layers of various types (contrasting structures, moving interior layers: fronts), which leads to the need to develop new asymptotic methods in order to study them both formally and rigorously. A general scheme for a rigorous study of contrast structures in singularly perturbed problems for partial differential equations, based on the use of the asymptotic method of differential inequalities, is presented and illustrated on relevant problems. The main achievements of this line of studies of partial differential equations are reflected, and the key directions of its development are indicated.

The presence of noise in images is unavoidable.It may be formed by image formation process,image recording,image transmission etc.These random distortions make it difficult to perform any required image processing task.High quality noise free remote sensing images are necessary for various applications.Therefore, removal of noise is necessary. The purpose of image denoising is to preserve edges as far as possible while removing noise, making the resulting images approximate the ideal image.This paper proposes a method for denoising remote sensing images using a combination of second order and fourth order partial differential equations.The advantages of both second order and fourth order partial differential equations are utilized here.The image is denoised using second order partial differential equations , fourth order partial differential equations and the combination of both.A comparison of the three methods is also presented.The proposed algorithm smooth out more noise and conserve more detail in the denoising process.

The nonlocal boundary-value problem for a third order partial differential equation in a Hilbert space H with a self-adjoint positive definite operator A is considered. A stable three-step difference scheme for the approximate solution of the problem is presented. The main theorem on stability of this difference scheme is established. In applications, the stability estimates for the solution of difference schemes of the approximate solution of three nonlocal boundary value problems for third order partial differential equations are obtained. Numerical results for oneand two-dimensional third order partial differential equations are provided.

The mathematical modeling of numerous real-world applications results in hierarchical optimization problems with two decision makers where at least one of them has to solve an optimal control problem of ordinary or partial differential equations. Such models are referred to as bilevel optimal control problems. Here, we first review some different features of bilevel optimal control including important applications, existence results, solution approaches, and optimality conditions. Afterwards, we focus on a specific problem class where parameters appearing in the objective functional of an optimal control problem of partial differential equations have to be reconstructed. After verifying the existence of solutions, necessary optimality conditions are derived by exploiting the optimal value function of the underlying parametric optimal control problem in the context of a relaxation approach.

In this paper, the proposed modification to Adomian decomposition method (ADM) is introduced. In addition, a new modification to solve singular and nonsingular initial-value problems (IVPs) in first order partial differential equations (PDEs) is investigated.
 Then, a discussion of how ADM can be used to solve singular and nonsingular IVPs is produced. Finally, some example are given to demonstrate the applicability of the results of paper.
 Key words; Adomian decomposition method; Nonlinear first-order PDE; Singular and nonsingular first-order partial differential equations.
 1 Introduction
 Singular nonlinear partial differential equations (PDEs) arise in various physical phe- nomena in applied sciences and engineering from such areas as fluid mechanics and heat-

In this paper, we propose a local and contextual controlled (LCC) fourth order partial differential equation (PDE) method for noise removal. First, a region based fourth order PDE is proposed, which conductance coefficients are chosen adaptively in terms of the domain type. Thus, the proposed method can preserve the advantages of forth order PDE and avoid leaving isolated black and white speckles. Furthermore, the region based fourth order PDE and two discontinuity measures are incorporated into a LCC fourth order PDE, where the joint use of the two discontinuity measures leads to a complementary effect for edge preservation. The proposed LCC fourth order PDE is tested on two typical images, including texture image and natural scene image, which show the superiority of the proposed method.

In this work, we investigate a unique solvability of a direct and inverse source problem for a time-fractional partial differential equation with the Caputo and Bessel operators. Using spectral expansion method, we give explicit forms of solutions to formulated problems in terms of multinomial Mittag-Leffler and first kind Bessel functions.

New index transforms are investigated, which contain as the kernel products of the Bessel and modified Bessel functions. Mapping properties and invertibility in Lebesgue spaces are studied for these operators. Relationships with the Kontorovich–Lebedev and Fourier cosine transforms are established. Inversion theorems are proved. As an application, a solution of the initial value problem for the fourth order partial differential equation, involving the Laplacian is presented.

Integral Equations for Nonlinear Problems in Partial Differential Equations

Many authors used different techniques for image restoration. Variational approaches are mainly used for this purpose. For example, the Mumfort-Shah is a model type for the free discontinuity in the restoration and segmentation of images. The main characteristic of this model is the isotropic diffusion, minimizing the length of the boundaries, while the classical models due to Rudin-Osher-Fatmi, minimize the total variation of the longest curves in the images. In this work we propose a new partial differential equation of order four that can be used for image restoration. This proposed model is a combination of the fast growth with respect to low curvature and the slow growth when the gradient is large. Numerical results are presented for both second and fourth order partial differential equations.

Modelica is a new object-oriented multi-domain modelling and simulation language used for solving large, complex and heterogeneous physical systems with differential algebraic equations (DAEs). Presently, there is no simulation support available in Modelica for solving partial differential equations (PDEs) problems. In this paper, based on the object-oriented idea, we present a method of consistent representation of PDE and DAE problems. The description of PDE problems in Modelica is introduced. Then the PDE model is transferred into DAE model with the method of lines, and is solved in MWorks simulation platform based on Modelica. To illustrate the application of our method, a heat conduction problem is modelled and simulated in Modelica.

Analytical C2 smooth blending surfaces