scholarly journals Inverse Problem for a Partial Differential Equation with Gerasimov–Caputo-Type Operator and Degeneration

2021 ◽  
Vol 5 (2) ◽  
pp. 58
Author(s):  
Tursun K. Yuldashev ◽  
Bakhtiyar J. Kadirkulov
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Fourier Series ◽  
Three Dimensional ◽  
Integral Form ◽  
Schwarz Inequality ◽  
Rectangular Domain ◽  
Countable System ◽  
Partial Differential ◽  
Fractional Differential

In the three-dimensional open rectangular domain, the problem of the identification of the redefinition function for a partial differential equation with Gerasimov–Caputo-type fractional operator, degeneration, and integral form condition is considered in the case of the 0<α≤1 order. A positive parameter is present in the mixed derivatives. The solution of this fractional differential equation is studied in the class of regular functions. The Fourier series method is used, and a countable system of ordinary fractional differential equations with degeneration is obtained. The presentation for the redefinition function is obtained using a given additional condition. Using the Cauchy–Schwarz inequality and the Bessel inequality, the absolute and uniform convergence of the obtained Fourier series is proven.

Adaptive Finite Element Method for Solving Poisson Partial Differential Equation

2021 ◽  
Vol 4 (1) ◽  
pp. 1-18
Author(s):  
Gokul KC ◽  
Ram Prasad Dulal
Keyword(s):  
Differential Equation ◽  
Finite Element Method ◽  
Partial Differential Equation ◽  
Finite Element ◽  
Poisson Equation ◽  
Rectangular Domain ◽  
Adaptive Finite Element Method ◽  
Adaptive Finite Element ◽  
Partial Differential ◽  
Element Method

Poisson equation is an elliptic partial differential equation, a generalization of Laplace equation. Finite element method is a widely used method for numerically solving partial differential equations. Adaptive finite element method distributes more mesh nodes around the area where singularity of the solution happens. In this paper, Poisson equation is solved using finite element method in a rectangular domain with Dirichlet and Neumann boundary conditions. Posteriori error analysis is used to mark the refinement area of the mesh. Dorfler adaptive algorithm is used to refine the marked mesh. The obtained results are compared with exact solutions and displayed graphically.

Three-Dimensional Elastic Solution of a Transversely Isotropic Functionally Graded Rectangular Plate

2012 ◽  
Vol 591-593 ◽  
pp. 2655-2660 ◽  
Author(s):  
Guo Jun Nie ◽  
Zhao Yang Feng ◽  
Jun Tao Shi ◽  
Ying Ya Lu ◽  
Zheng Zhong
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Rectangular Plate ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Functionally Graded ◽  
Elastic Solution ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Present Solution

Three-dimensional elastic solution of a simply supported, transversely isotropic functionally graded rectangular plate is presented in this paper. Suppose that all elastic coefficients of the material have the same power-law dependence on the thickness coordinate. By introducing two new displacement functions, three equations of equilibrium in terms of displacements are reduced to two uncoupled partial differential equations. Exact solution for a second-order partial differential equation expressed by one of displacement functions is obtained and analytical solution for another fourth-order partial differential equation expressed by another displacement function is found by employing the Frobenius method. The validity of the present solution is first investigated. And the effect of the gradation of material properties on the mechanical behavior of the plate is studied through numerical examples.

A new partial differential equation for image inpainting

2021 ◽  
Vol 39 (3) ◽  
pp. 137-155
Author(s):  
Mounder Benseghir ◽  
Fatma Zohra Nouri ◽  
Pierre Clovis Tauber
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Applied Mathematics ◽  
Three Dimensional ◽  
Image Inpainting ◽  
Complex Geometries ◽  
Partial Differential ◽  
Nonlinear Structure ◽  
Grey Scale ◽  
Nonlinear High Order

A considerable interest in the inpainting problem have attracted many researchers in applied mathematics community. In fact in the last decade, nonlinear high order partial dierential equations have payed a central role in high quality inpainting developments. In this paper, we propose a technique for inpainting that combines an anisotropic diusion process with an edge-corner enhancing shock ltering. This technique makes use of a partial differential equation that is based on a nonlinear structure tensor which increases the accuracy and robustness of the coupled diusion and shock ltering. A methodology of partition and adjustment is used to estimate the contrast parameters that control the strength of the diffusivity functions. We focus on restoring large missing regions in grey scale images containing complex geometries parts. Our model is extended to a three dimensional case, where numerical experimentations were carried out on lling brain multiple sclerosis lesions in medical images. The efficiency and the competitiveness of the proposed algorithm is numerically compared to other approaches on both synthetic and real images.

scholarly journals Boundary Value Problem for Weak Nonlinear Partial Differential Equations of Mixed Type with Fractional Hilfer Operator

Axioms ◽  
2020 ◽  
Vol 9 (2) ◽  
pp. 68 ◽  
Author(s):  
Tursun K. Yuldashev ◽  
Bakhtiyor J. Kadirkulov
Keyword(s):  
Differential Equation ◽  
Fourier Series ◽  
Boundary Value Problem ◽  
Spectral Parameter ◽  
Mixed Type ◽  
Boundary Value ◽  
Nonlinear Partial Differential Equation ◽  
Rectangular Domain ◽  
Partial Differential ◽  
Negative Part

In this paper, we consider a boundary value problem for a nonlinear partial differential equation of mixed type with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. With respect to the first variable, this equation is a nonlinear fractional differential equation in the positive part of the considering segment and is a second-order nonlinear differential equation with spectral parameter in the negative part of this segment. Using the Fourier series method, the solutions of nonlinear boundary value problems are constructed in the form of a Fourier series. Theorems on the existence and uniqueness of the classical solution of the problem are proved for regular values of the spectral parameter. For irregular values of the spectral parameter, an infinite number of solutions of the mixed equation in the form of a Fourier series are constructed.

Three-dimensional invisible cloaks with arbitrary shapes based on partial differential equation

2010 ◽  
Vol 216 (2) ◽  
pp. 426-430 ◽  
Author(s):  
Xinhua Wang ◽  
Shaobo Qu ◽  
Zhuo Xu ◽  
Hua Ma ◽  
Jiafu Wang ◽  
...  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Three Dimensional ◽  
Partial Differential ◽  
Arbitrary Shapes

scholarly journals Exploring Capabilities within ForTrilinos by Solving the 3D Burgers Equation

2012 ◽  
Vol 20 (3) ◽  
pp. 275-292 ◽  
Author(s):  
Karla Morris ◽  
Damian W.I. Rouson ◽  
M. Nicole Lemaster ◽  
Salvatore Filippone
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Message Passing ◽  
Message Passing Interface ◽  
Mathematical Formulation ◽  
Three Dimensional ◽  
Abstract Data Type ◽  
Partial Differential ◽  
Sparse Linear Solvers ◽  
Set Up

We present the first three-dimensional, partial differential equation solver to be built atop the recently released, open-source ForTrilinos package (http://trilinos.sandia.gov/packages/fortrilinos). ForTrilinos currently provides portable, object-oriented Fortran 2003 interfaces to the C++ packages Epetra, AztecOO and Pliris in the Trilinos library and framework [ACM Trans. Math. Softw.31(3) (2005), 397–423]. Epetra provides distributed matrix and vector storage and basic linear algebra calculations. Pliris provides direct solvers for dense linear systems. AztecOO provides iterative sparse linear solvers. We demonstrate how to build a parallel application that encapsulates the Message Passing Interface (MPI) without requiring the user to make direct calls to MPI except for startup and shutdown. The presented example demonstrates the level of effort required to set up a high-order, finite-difference solution on a Cartesian grid. The example employs an abstract data type (ADT) calculus [Sci. Program.16(4) (2008), 329–339] that empowers programmers to write serial code that lower-level abstractions resolve into distributed-memory, parallel implementations. The ADT calculus uses compilable Fortran constructs that resemble the mathematical formulation of the partial differential equation of interest.

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