scholarly journals NECESSARY NON-LOCAL CONDITIONS FOR A DIFFUSION-WAVE EQUATION

2017 ◽  
Vol 20 (7) ◽  
pp. 45-59 ◽  
Author(s):  
M.O. Mamchuev
Keyword(s):  
Wave Equation ◽  
Fractional Derivative ◽  
Explicit Form ◽  
Existence And Uniqueness ◽  
Integral Operators ◽  
Rectangular Domain ◽  
Uniqueness Of Solution ◽  
Local Conditions ◽  
Diffusion Wave ◽  
Non Local

In this article, diffusion-wave equation with fractional derivative in Rieman- n-Liouville sense is investigated. Integral operators with the Write function in the kernel associated with the investigational equation are introduced. In terms of these operators necessary non-local conditions binding traces of solution and its derivatives on the boundary of a rectangular domain are found. Necessary non-local conditions for the wave are obtained by using the limiting properties of Write function. By using the integral operator’s properties the theorem of existence and uniqueness of solution of the problem with integral Samarski’s condition for the diffusion-wave equation is proved. The solution is obtained in explicit form.

scholarly journals Solvability of BVPs for the Parabolic-Hyperbolic Equation with Non-linear Loaded Term

2021 ◽  
pp. 133-145
Author(s):  
Obidjon Kh. Abdullaev
Keyword(s):  
Hyperbolic Equation ◽  
Existence And Uniqueness ◽  
Type Equation ◽  
Existence Of Solution ◽  
Hyperbolic Type ◽  
Uniqueness Of Solution ◽  
Method Of Integral Equations ◽  
Non Local ◽  
Integral Energy ◽  
Loaded Term

This work is devoted to prove the existence and uniqueness of solution of BVP with non-local assumptions on the boundary and integral gluing conditions for the parabolic-hyperbolic type equation involving Caputo derivatives. Using the method of integral energy, the uniqueness of solution have been proved. Existence of solution was proved by the method of integral equations

scholarly journals Existence, uniqueness and stability results for semilinear integrodifferential non-local evolution equations with Random impulse

Filomat ◽  
2018 ◽  
Vol 32 (19) ◽  
pp. 6615-6626
Author(s):  
B. Radhakrishnan ◽  
M. Tamilarasi ◽  
P. Anukokila
Keyword(s):  
Hilbert Spaces ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Semigroup Theory ◽  
Local Conditions ◽  
Fixed Point Approach ◽  
Non Local ◽  
The Stability ◽  
Random Impulse ◽  
Stability Results

In this paper, authors investigated the existence and uniqueness of random impulsive semilinear integrodifferential evolution equations with non-local conditions in Hilbert spaces. Also the stability results for the same evolution equation has been studied. The results are derived by using the semigroup theory and fixed point approach. An application is provided to illustrate the theory.

scholarly journals DIRCHLET PROBLEM FOR A NONLOCAL WAVE EQUATION WITH RIEMANN-LIOUVILLE DERIVATIVE

2019 ◽  
pp. 6-11
Author(s):  
О.Х. Масаева
Keyword(s):  
Wave Equation ◽  
Dirichlet Problem ◽  
Fractional Derivative ◽  
Existence And Uniqueness ◽  
Liouville Derivative ◽  
Uniqueness Of The Solution

Доказано существование и единственность решения задачи Дирихле для уравнения второго порядка с дробной производной. Исследуемое уравнение переходит в волновое уравнение при целом значении порядка дробной производной The existence and uniqueness of the solution to Dirichlet problem for a secondorder equation with a fractional derivative is proved. The equation under study is a wave equation for a integer value of the order of the fractional derivative.

scholarly journals Time-Fractional Diffusion-Wave Equation with Mass Absorption in a Sphere under Harmonic Impact

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 433 ◽  
Author(s):  
Bohdan Datsko ◽  
Igor Podlubny ◽  
Yuriy Povstenko
Keyword(s):  
Fourier Transform ◽  
Wave Equation ◽  
Fractional Derivative ◽  
Graphical Representation ◽  
Fractional Diffusion ◽  
Diffusion Wave ◽  
Mass Absorption ◽  
The Laplace Transform ◽  
Time Fractional Diffusion Equation ◽  
Time Fractional Derivative

The time-fractional diffusion equation with mass absorption in a sphere is considered under harmonic impact on the surface of a sphere. The Caputo time-fractional derivative is used. The Laplace transform with respect to time and the finite sin-Fourier transform with respect to the spatial coordinate are employed. A graphical representation of the obtained analytical solution for different sets of the parameters including the order of fractional derivative is given.

scholarly journals Analysis of two Legendre spectral approximations for the variable-coefficient fractional diffusion-wave equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Wenping Chen ◽  
Shujuan Lü ◽  
Hu Chen ◽  
Lihua Jiang
Keyword(s):  
Wave Equation ◽  
Fractional Derivative ◽  
Variable Coefficient ◽  
Equivalent Form ◽  
Fractional Diffusion ◽  
Stability And Convergence ◽  
Diffusion Wave ◽  
Fully Discrete ◽  
Spectral Approximations ◽  
Time Fractional Derivative

Abstract In this paper, we solve the variable-coefficient fractional diffusion-wave equation in a bounded domain by the Legendre spectral method. The time fractional derivative is in the Caputo sense of order $\gamma \in (1,2)$ γ ∈ ( 1 , 2 ) . We propose two fully discrete schemes based on finite difference in temporal and Legendre spectral approximations in spatial discretization. For the first scheme, we discretize the time fractional derivative directly by the $L_{1}$ L 1 approximation coupled with the Crank–Nicolson technique. For the second scheme, we transform the equation into an equivalent form with respect to the Riemann–Liouville fractional integral operator. We give a rigorous analysis of the stability and convergence of the two fully discrete schemes. Numerical examples are carried out to verify the theoretical results.

Time distributed-order diffusion-wave equation. II. Applications of Laplace and Fourier transformations

2009 ◽  
Vol 465 (2106) ◽  
pp. 1893-1917 ◽  
Author(s):  
Teodor M. Atanackovic ◽  
Stevan Pilipovic ◽  
Dusan Zorica
Keyword(s):  
Wave Equation ◽  
Fractional Derivative ◽  
Fractional Diffusion ◽  
Diffusion Wave ◽  
The Maximum Principle ◽  
Dimensional Diffusion ◽  
First Case ◽  
Special Cases ◽  
Telegraph Equations ◽  
Distributed Order

A Cauchy problem for a time distributed-order multi-dimensional diffusion-wave equation containing a forcing term is reinterpreted in the space of tempered distributions, and a distributional diffusion-wave equation is obtained. The distributional equation is solved in the general case of weight function (or distribution). Solutions are given in terms of solution kernels (Green's functions), which are studied separately for two cases. The first case is when the order of the fractional derivative is in the interval [0, 1], while, in the second case, the order of the fractional derivative is in the interval [0, 2]. Solutions of fractional diffusion-wave and fractional telegraph equations are obtained as special cases. Numerical experiments are also performed. An analogue of the maximum principle is also presented.

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