关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

1. Metric, Banach, and Hilbert Spaces.- 2. Linear Operators on Banach Spaces.- 3. Almost Periodic Functions.- 4. Almost Automorphic Functions.- 5. Pseudo-Almost Periodic Functions.- 6. Pseudo-Almost Automorphic Functions.- 7. Existence Results for Some Second-Order Differential Equations.- 8. Existence Results to Some Integrodifferential Equations.- 9. Existence of C(m)-Pseudo-Almost Automorphic Solutions.- 10. Pseudo-Almost Periodic Solutions to Some Third-Order Differential Equations.- 11. Pseudo-Almost Automorphic Solutions to Some Sobolev-Type Equations.- 12. Stability Results for Some Higher-Order Difference Equations.- 13. Appendix A.- References.- Index.

https://www.springer.com/cda/content/document/productFlyer/productFlyer_978-3-319-00848-6.pdf?SGWID=0-0-1297-175260342-0

Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces

When the structures are analyzed by classical method, several governing differential equations are needed if a beam is loaded by many loads But if delta function and its derivatives as generalized functions are used, one governing differential equation is enough In Lw(x) = q(x), where L is a linear differential operator, the particular solution Wp(x) can be obtained as a integral from using the method of variation of parameters Since the external load term, q(x), can be described by using these generalized functions, the filtration property of delta function in a integral makes the form of Wp(x) simpler one The usage of these generalized functions is shown in several sample cases and we can see the convenience of the generalized functions

구조해석에서의 Generalized Functions의 응용

Local and global existence of mild solution to an impulsive fractional functional integro-differential equation with nonlocal condition

Abstract This paper concerns the dissipativity and stability of the Caputo nonlinear fractional functional differential equations (F-FDEs) with order 0 < α < 1. The fractional generalization of the Halanay-type inequality is proposed, which plays a central role in studies of stability and dissipativity of F-FDEs. Then the dissipativity and the absorbing set are derived under almost the same assumptions as the classical integer-order functional differential equations (FDEs). The asymptotic stability of F-FDEs are also proved under the one-sided Lipschitz conditions. Those extend the corresponding properties from integer-order FDEs to the Caputo fractional ones. The results can also be directly applied to some special cases of fractional nonlinear equations, such as the fractional delay differential equations (F-DDEs), fractional integro-differential equations (F-IDEs) and fractional delay integro-differential equations (F-DIDEs). The fractional Adams-Bashforth-Moulton algorithm is employed to simulate the F-FDEs, and several numerical examples are given to illustrate the theoretical results.

Dissipativity and Stability Analysis for Fractional Functional Differential Equations

In the present work, we investigate the numerical solution of time-fractional telegraph equation by a local meshless method. The fractional-order derivative is defined in the Caputo’s sense. The time semi-discretization was carried out using finite difference method followed by radial basis function-based spatial discretization. The theoretical convergence analysis and stability analysis of time semi-discrete scheme are also proved. Several test problems with regular and irregular domains with uniform and non-uniform points are considered. To demonstrate the accuracy and efficiency of the proposed method, we compared the analytical and numerical solution of the proposed problem.

A local meshless method to approximate the time-fractional telegraph equation

We present a numerical approximate solution to Sobolev-type dif- ferential equation subject to nonlocal initial boundary conditions. A Laplace transform method is described for the solution of considered equation. Fol- lowing Laplace transform of the original problem, an appropriate method of solving dierential equations is used to solve the resultant time-independent modified equation and solution is inverted numerically back into the time do- main. Numerical results are provided to show the accuracy of the proposed method.

https://ejde.math.txstate.edu/conf-proc/19/d3/dubey.pdf

Numerical solution for nonlocal sobolev-type differential equations

In this paper, we discuss the solutions to nonlocal initial value problems of fractional order functional differential equations in a Banach space. In particular, we prove the existence and uniqueness of mild and classical solutions assuming that −A generates a resolvent operator family and nonlinear part is a Lipschitz continuous function. We also investigate the global existence of the solution. At the end, a fractional order partial differential equation is given to illustrate the obtained abstract results.

Solutions to fractional functional differential equations with nonlocal conditions

This article deals with the analytical study of propagation of static wall profiles in ferromagnetic nanowires under the effect of spin–orbit Rashba field. We consider the governing dynamics for the evolution of magnetization inside the ferromagnetic material as an extended version of Landau–Lifshitz–Gilbert–Slonczewski equation of micromagnetism. It comprises the nonlinear dissipation factors like dry-friction and viscous. We establish the threshold and Walker-type breakdown estimates of the external sources in the steady-regime and also illustrate the obtained results numerically.

On Dynamics of Current-Induced Static Wall Profiles in Ferromagnetic Nanowires Governed by the Rashba Field

Abstract We investigate the stability features of steady-states of a two-dimensional system of ferromagnetic nanowires. We constitute a system with the finite number of nanowires arranged on the ( e → 1 , e → 2 ) {(\\vec{e}_{1},\\vec{e}_{2})} plane, where ( e → 1 , e → 2 , e → 3 ) {(\\vec{e}_{1},\\vec{e}_{2},\\vec{e}_{3})} is the canonical basis of ℝ 3 {\\mathbb{R}^{3}} . We consider two cases: in the first case, each nanowire is considered to be of infinite length, whereas in the second case, we deal with finite length nanowires to design the system. In both cases, we establish a sufficient condition under which these steady-states are shown to be exponentially stable.

Shruti Dubey

Papers

A local meshless method to approximate the time-fractional telegraph equation

Numerical solution for nonlocal sobolev-type differential equations

Solutions to fractional functional differential equations with nonlocal conditions

On Dynamics of Current-Induced Static Wall Profiles in Ferromagnetic Nanowires Governed by the Rashba Field

On the stability of steady-states of a two-dimensional system of ferromagnetic nanowires