Investigation Of Partial Differential Equations Research Articles

Method of directly defining the inverse mapping for nonlinear fuzzy heat-like partial differential equations

Natural phenomena or physical systems can be described using Partial Differential Equations (PDEs), such as wave equations, heat equations, Poisson’s equation, and so on. Consequently, investigations of PDEs have become one of the key areas of modern mathematical analyses, attracting a lot of attention. Many authors have recently expressed an interest in researching the theoretical framework of fuzzy Initial Value Problems (IVPs). The Method of Directly Defining the inverse Mapping (MDDiM) was effectively employed in this research to obtain the second-order approximate fuzzy solution of heatlike equations in one and two dimensions, and the results were compared with exact solutions. In each illustrated example, all the results achieved using Maple 16 were graphically depicted. This is the first time MDDiM was utilized to solve nonlinear Fuzzy Partial Differential Equations (FPDEs).

An adaptive stochastic investigation of partial differential equations using wavelet collocation generalized polynomial chaos method

Starting from offering good solution to differential equations to capturing the nonlinearity, wavelets have gained more and more attention even for handling the uncertainties present in a physical system. In this paper, wavelet collocation approach has been united with the polynomial chaos expansion for numerically solving the stochastic partial differential equations. This approach is associated with the concept of autocorrelation functions of compactly supported Daubechies scaling functions. First of all, we make use of linear B-spline basis in generalized polynomial chaos. Then, in order to separate the randomness, Galerkin projection is executed on uncertain data and solution variables. After that, connection coefficients are calculated using Daubechies wavelets for approximating the differential operators. Moreover, fast algorithms are known to speed up the numerical scheme, so, we have executed a class of fast algorithms, on the basis of fast wavelet transform. Also, in order to handle the condition number, we have used a good diagonal preconditioning technique which makes the condition number of the matrices bounded. We have also executed an adaptive scheme which will focus on capturing the essential features of the solution by modifying the grids as well as reducing the CPU time. The scheme has been tested along with adaptivity on three stochastic partial differential equations with uncertain initial data and the results obtained from the proposed method are quite promising.

Partial fractional differential equations and some of their applications

Our paper is devoted to investigation of partial differential equations of fractional order. We give a historical survey of results in this field basically concerning differential equations with Riemann–Liouville and Caputo partial fractional derivatives. We pay a special attention to application of the method of Fourier, Laplace and Mellin integral transforms to study partial fractional differential equations. We present recent results on explicit solutions of Cauchy-type and Cauchy problems for model homogeneous partial differential equations with Riemann–Liouville and Caputo partial fractional derivatives generalizing the classical heat and wave equations. Explicit solutions of the above problems are given in terms of the Mittag–Leffler function, and of the so-called H-function and its special cases such as the Wright and generalized Wright functions.

A simplifying transformation for the Laplace-Beltrami operator in curvilinear coordinates

In a curvilinear coordinate system with metric tensor G, the Laplace-Beltrami operator ▿ 2 expresses the Laplacian in terms of partial derivatives with respect to the coordinates. This paper describes a simplifying transformation, useful in curvilinear coordinate systems with a nondiagonal G, where the mixed partial derivative terms are problematic. G is expressed as the matrix multiple G = F G ̌ , where G ̌ is diagonal. Using the transformation χ = f 1 4 ψ , where f = det( F), the result ▿ 2ψ = f − 1 4 (▿ 0 2χ+K 0χ+U 0χ) is obtained, where ▿ 0 2 is the Laplacian in a “straightened-out” coordinate system, perturbed by differential and multiplication operators K 0 and U 0. This allows the investigation of partial differential equations in complicated geometries by perturbation methods in simpler geometries. An illustrative example is given.

Global analysis of the phase portrait for the Kuramoto-Sivashinsky equation

The global behavior of the Kuramoto-Sivashinsky equation is studied. The existence of an absorbing ball in every Sobolev norm is proved. The transition of energy from low modes to high ones is observed. An upper estimate for the Hausdorff dimension of the attractor is given. The main tool is to use the methods of the theory of ordinary differential equations in the investigation of partial differential equations.