scholarly journals Two-Dimensional Advection–Diffusion Process with Memory and Concentrated Source

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 879 ◽  
Author(s):  
Najma Ahmed ◽  
Nehad Ali Shah ◽  
Dumitru Vieru
Keyword(s):  
Differential Equation ◽  
Fourier Transforms ◽  
Diffusion Processes ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Two Dimensional ◽  
Symmetry Center ◽  
Advection Diffusion ◽  
Initial Boundary Value ◽  
With Memory

Two-dimensional advection–diffusion processes with memory and a source concentrated in the symmetry center of the domain have been investigated. The differential equation of the studied model is a fractional differential equation with short-tail memory (a differential equation with Caputo–Fabrizio time-fractional derivatives). An analytical solution of the initial-boundary value problem has been determined by employing the Laplace transform and double sine-Fourier transforms. A numerical solution of the studied problem has been determined using finite difference approximations. Numerical simulations for both solutions have been carried out using the software Mathcad.

scholarly journals Construction of an Explicit Solution of a Time-Fractional Multidimensional Differential Equation

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2052
Author(s):  
Murat A. Sultanov ◽  
Durdimurod K. Durdiev ◽  
Askar A. Rahmonov
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Explicit Solution ◽  
Fourier Transforms ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Laplace And Fourier Transforms ◽  
Memory Kernel ◽  
Fractional Differential ◽  
Initial Boundary Value

In this work, an explicit solution of the initial-boundary value problem for a multidimensional time-fractional differential equation is constructed. The possibility of obtaining this equation from an integro-differential wave equation with a Mittag–Leffler–type memory kernel is shown. An explicit solution to the problem under consideration is obtained using the Laplace and Fourier transforms, the properties of the Fox H-functions and the convolution theorem.

scholarly journals GENERALIZED SOLUTIONS FOR THE EULER–BERNOULLI MODEL WITH ZENER VISCOELASTIC FOUNDATIONS AND DISTRIBUTIONAL FORCES

2013 ◽  
Vol 11 (02) ◽  
pp. 1350017 ◽  
Author(s):  
GÜNTHER HÖRMANN ◽  
SANJA KONJIK ◽  
LJUBICA OPARNICA
Keyword(s):  
Differential Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variational Problems ◽  
Generalized Solutions ◽  
Beam Model ◽  
Order Partial Differential Equation ◽  
Viscoelastic Foundation ◽  
Regularization Techniques ◽  
Initial Boundary Value

We study the initial-boundary value problem for an Euler–Bernoulli beam model with discontinuous bending stiffness laying on a viscoelastic foundation and subjected to an axial force and an external load both of Dirac-type. The corresponding model equation is a fourth-order partial differential equation and involves discontinuous and distributional coefficients as well as a distributional right-hand side. Moreover the viscoelastic foundation is of Zener-type and described by a fractional differential equation with respect to time. We show how functional analytic methods for abstract variational problems can be applied in combination with regularization techniques to prove existence and uniqueness of generalized solutions.

scholarly journals Well-posedness of time-fractional advection-diffusion-reaction equations

2019 ◽  
Vol 22 (4) ◽  
pp. 918-944 ◽  
Author(s):  
William McLean ◽  
Kassem Mustapha ◽  
Raed Ali ◽  
Omar Knio
Keyword(s):  
Linear Time ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Diffusion Reaction ◽  
Fractional Integrals ◽  
Well Posedness ◽  
Advection Diffusion ◽  
Initial Boundary Value ◽  
Reaction Equations ◽  
Advection Diffusion Reaction Equations

Abstract We establish the well-posedness of an initial-boundary value problem for a general class of linear time-fractional, advection-diffusion-reaction equations, allowing space- and time-dependent coefficients as well as initial data that may have low regularity. Our analysis relies on novel energy methods in combination with a fractional Gronwall inequality and properties of fractional integrals.

scholarly journals Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions

2017 ◽  
Vol 15 (1) ◽  
pp. 404-419 ◽  
Author(s):  
Yuriy Golovaty ◽  
Volodymyr Flyud
Keyword(s):  
Differential Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Singularly Perturbed ◽  
Hyperbolic Problems ◽  
Metric Graphs ◽  
Asymptotics Of Solutions ◽  
Layer Method ◽  
Boundary Layer Method ◽  
Initial Boundary Value

Abstract We are interested in the evolution phenomena on star-like networks composed of several branches which vary considerably in physical properties. The initial boundary value problem for singularly perturbed hyperbolic differential equation on a metric graph is studied. The hyperbolic equation becomes degenerate on a part of the graph as a small parameter goes to zero. In addition, the rates of degeneration may differ in different edges of the graph. Using the boundary layer method the complete asymptotic expansions of solutions are constructed and justified.

scholarly journals Exact variational representations for the solution of the simple parabolic equation

1971 ◽  
Vol 5 (3) ◽  
pp. 305-314
Author(s):  
R.S. Anderssen
Keyword(s):  
Differential Equation ◽  
Fourier Series ◽  
Parabolic Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Parabolic Differential Equation ◽  
Initial Boundary Value Problems ◽  
Series Representations ◽  
Initial Boundary Value

By constructing a special set of A-orthonormal functions, it is shown that, under certain smoothness assumptions, the variational and Fourier series representations for the solution of first initial boundary value problems for the simple parabolic differential equation coincide. This result is then extended in order to construct a variational representation for the solution of a very general first initial boundary value problem for this equation.

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