scholarly journals Theoretical and Numerical Aspect of Fractional Differential Equations with Purely Integral Conditions

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1987
Author(s):  
Saadoune Brahimi ◽  
Ahcene Merad ◽  
Adem Kılıçman
Keyword(s):  
Boundary Conditions ◽  
A Priori ◽  
A Priori Estimate ◽  
Numerical Tests ◽  
Advection Diffusion Equation ◽  
Nonhomogeneous Boundary ◽  
Estimate Method ◽  
Fractional Differential ◽  
The Given ◽  
Range Density

In this paper, we are interested in the study of a Caputo time fractional advection–diffusion equation with nonhomogeneous boundary conditions of integral types ∫01vx,tdx and ∫01xnvx,tdx. The existence and uniqueness of the given problem’s solution is proved using the method of the energy inequalities known as the “a priori estimate” method relying on the range density of the operator generated by the considered problem. The approximate solution for this problem with these new kinds of boundary conditions is established by using a combination of the finite difference method and the numerical integration. Finally, we give some numerical tests to illustrate the usefulness of the obtained results.

FINITE-DIFFERENCE METHODS FOR PROBLEM OF CONJUGATION OF HYPERBOLIC AND PARABOLIC EQUATIONS

2000 ◽  
Vol 10 (03) ◽  
pp. 361-377 ◽  
Author(s):  
ALEXANDER A. SAMARSKII ◽  
VICTOR I. KORZYUK ◽  
SERGEY V. LEMESHEVSKY ◽  
PETR P. MATUS
Keyword(s):  
Difference Scheme ◽  
Parabolic Equations ◽  
A Priori ◽  
Finite Difference Methods ◽  
A Priori Estimate ◽  
Stability And Convergence ◽  
Priori Estimate ◽  
Order Of Accuracy ◽  
Spatial Operator ◽  
The Given

A problem of conjugation of hyperbolic and parabolic equations in domain with moving boundaries is considered. Existence and uniqueness of a strong solution of the given problem are proved. A priori estimate for operator-difference scheme with non-self-adjoint spatial operator is obtain. Homogeneous difference scheme with constant weights for the conjugation problem is constructed. Moreover, consistency conditions are approximated with the second-order of accuracy with respect to spatial variables. Stability and convergence of the suggested scheme are investigated.

Pullback Exponential Attractors for Nonautonomous Reaction–Diffusion Equations

2015 ◽  
Vol 25 (05) ◽  
pp. 1550063
Author(s):  
Xingjie Yan ◽  
Wei Qi
Keyword(s):  
A Priori ◽  
Reaction Diffusion ◽  
A Priori Estimate ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Exponential Attractor ◽  
Exponential Attractors ◽  
Estimate Method ◽  
Necessary And Sufficient ◽  
Asymptotic A Priori Estimate

This paper presents a necessary and sufficient condition to prove the existence of the pullback exponential attractor. The asymptotic a priori estimate method is used to produce an abstract result on the existence of the pullback exponential attractor in a strong space without regularity. The established results are illustrated by applying them to the nonautonomous reaction–diffusion equations to prove the existence of the pullback exponential attractors in L2(Ω), [Formula: see text] and Lp(Ω)(p > 2) spaces.

scholarly journals On the solvability of a class of singular parabolic equations with nonlocal boundary conditions in nonclassical function spaces

2002 ◽  
Vol 30 (7) ◽  
pp. 435-447 ◽  
Author(s):  
Abdelfatah Bouziani
Keyword(s):  
Boundary Conditions ◽  
Parabolic Equations ◽  
A Priori ◽  
Nonlocal Boundary ◽  
A Priori Estimate ◽  
Nonlocal Boundary Conditions ◽  
Generalized Solution ◽  
Function Spaces ◽  
Singular Parabolic Equations ◽  
Abstract Formulation

The aim of this paper is to prove the existence, uniqueness, and continuous dependence upon the data of a generalized solution for certain singular parabolic equations with initial and nonlocal boundary conditions. The proof is based on an a priori estimate established in nonclassical function spaces, and on the density of the range of the operator corresponding to the abstract formulation of the considered problem.

scholarly journals A collocation method based on cubic trigonometric B-splines for the numerical simulation of the time-fractional diffusion equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Muhammad Yaseen ◽  
Muhammad Abbas ◽  
Muhammad Bilal Riaz
Keyword(s):  
Collocation Method ◽  
Fractional Derivatives ◽  
Fractional Diffusion ◽  
Biological Models ◽  
B Splines ◽  
Fractional Diffusion Equations ◽  
Numerical Tests ◽  
Symmetry Properties ◽  
Fractional Differential ◽  
The Given

AbstractFractional differential equations sufficiently depict the nature in view of the symmetry properties, which portray physical and biological models. In this paper, we present a proficient collocation method based on cubic trigonometric B-Splines (CuTBSs) for time-fractional diffusion equations (TFDEs). The methodology involves discretization of the Caputo time-fractional derivatives using the typical finite difference scheme with space derivatives approximated using CuTBSs. A stability analysis is performed to establish that the errors do not magnify. A convergence analysis is also performed The numerical solution is obtained as a piecewise sufficiently smooth continuous curve, so that the solution can be approximated at any point in the given domain. Numerical tests are efficiently performed to ensure the correctness and viability of the scheme, and the results contrast with those of some current numerical procedures. The comparison uncovers that the proposed scheme is very precise and successful.

scholarly journals Existence Results for Fractional Order Single-Valued and Multi-Valued Problems with Integro-Multistrip-Multipoint Boundary Conditions

2020 ◽  
Vol 4 (3) ◽  
pp. 31
Author(s):  
Sotiris K. Ntouyas ◽  
Bashir Ahmad ◽  
Ahmed Alsaedi
Keyword(s):  
Boundary Conditions ◽  
Existence Results ◽  
Inclusion Problem ◽  
Multipoint Boundary ◽  
New Class ◽  
Special Cases ◽  
Multipoint Boundary Conditions ◽  
Fractional Differential ◽  
Differential Equations And Inclusions ◽  
The Given

We study the existence of solutions for a new class of boundary value problems of arbitrary order fractional differential equations and inclusions, supplemented with integro-multistrip-multipoint boundary conditions. Suitable fixed point theorems are applied to prove some new existence results. The inclusion problem is discussed for convex valued as well as non-convex valued multi-valued map. Examples are also constructed to illustrate the main results. The results presented in this paper are not only new in the given configuration but also provide some interesting special cases.

scholarly journals PULLBACK ATTRACTORS FOR A NON-AUTONOMOUS SEMI-LINEAR DEGENERATE PARABOLIC EQUATION

2010 ◽  
Vol 52 (3) ◽  
pp. 537-554 ◽  
Author(s):  
CUNG THE ANH ◽  
TANG QUOC BAO
Keyword(s):  
Parabolic Equation ◽  
Degenerate Parabolic Equation ◽  
A Priori ◽  
Reaction Diffusion ◽  
A Priori Estimate ◽  
Reaction Diffusion Equations ◽  
Pullback Attractors ◽  
Estimate Method ◽  
Degenerate Parabolic ◽  
Linear Degenerate

AbstractIn this paper, using the asymptotic a priori estimate method, we prove the existence of pullback attractors for a non-autonomous semi-linear degenerate parabolic equation in an arbitrary domain, without restriction on the growth order of the polynomial type non-linearity and with a suitable exponential growth of the external force. The obtained results improve some recent ones for the non-autonomous reaction–diffusion equations.

scholarly journals Upper semicontinuity of attractors for nonclassical diffusion equations with arbitrary polynomial growth

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yongqin Xie ◽  
Jun Li ◽  
Kaixuan Zhu
Keyword(s):  
Polynomial Growth ◽  
A Priori ◽  
Upper Semicontinuity ◽  
A Priori Estimate ◽  
Global Attractors ◽  
Diffusion Equations ◽  
Nonlinear Anal ◽  
Estimate Method ◽  
Appl Math ◽  
Nonclassical Diffusion Equations

AbstractIn this paper, we mainly investigate upper semicontinuity and regularity of attractors for nonclassical diffusion equations with perturbed parameters ν and the nonlinear term f satisfying the polynomial growth of arbitrary order $p-1$ p − 1 ($p \geq 2$ p ≥ 2 ). We extend the asymptotic a priori estimate method (see (Wang et al. in Appl. Math. Comput. 240:51–61, 2014)) to verify asymptotic compactness and upper semicontinuity of a family of semigroups for autonomous dynamical systems (see Theorems 2.2 and 2.3). By using the new operator decomposition method, we construct asymptotic contractive function and obtain the upper semicontinuity for our problem, which generalizes the results obtained in (Wang et al. in Appl. Math. Comput. 240:51–61, 2014). In particular, the regularity of global attractors is obtained, which extends and improves some results in (Xie et al. in J. Funct. Spaces 2016:5340489, 2016; Xie et al. in Nonlinear Anal. 31:23–37, 2016).

scholarly journals Fractional-Order Integro-Differential Multivalued Problems with Fixed and Nonlocal Anti-Periodic Boundary Conditions

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1774 ◽  
Author(s):  
Ahmed Alsaedi ◽  
Ravi P. Agarwal ◽  
Sotiris K. Ntouyas ◽  
Bashir Ahmad
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Fixed Point Theory ◽  
Periodic Boundary Conditions ◽  
Point Theory ◽  
Caputo Fractional Derivatives ◽  
New Class ◽  
Fractional Differential ◽  
The Given ◽  
Derivatives Of

This paper studies a new class of fractional differential inclusions involving two Caputo fractional derivatives of different orders and a Riemann–Liouville type integral nonlinearity, supplemented with a combination of fixed and nonlocal (dual) anti-periodic boundary conditions. The existence results for the given problem are obtained for convex and non-convex cases of the multi-valued map by applying the standard tools of the fixed point theory. Examples illustrating the obtained results are presented.

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