Now showing 1 - 4 of 4
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    Strict Hölder regularity for fractional order abstract degenerate differential equations
    (01-01-2022)
    Alam, Md Mansur
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    In this paper, we first characterize the behaviour of fractional resolvent families on the real interpolation spaces (X,D(A))θ,p,θ∈(0,1),p∈[1,∞]. Second, we establish strict Hölder regularity in space and time to an abstract degenerate fractional order differential equations. We also provide an application to illustrate our results.
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    Mild Solutions of Time Fractional Navier-Stokes Equations Driven by Finite Delayed External Forces
    (01-04-2022)
    Alam, Md Mansur
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    In this work, we consider time-fractional Navier-Stokes equations (NSE) with the external forces involving finite delay. Equations are considered on a bounded domain Ω ⊂ R3 having sufficiently smooth boundary. We transform the system of equations (NSE) to an abstract Cauchy problem and then investigate local existence and uniqueness of the mild solutions for the initial datum (Formula Presented), where r > 0 and A is the Stokes operator. With some suitable condition on initial datum we establish the global continuation and regularity of the mild solutions.We use semigroup theory, tools of fractional calculus and Banach contraction mapping principle to establish our results
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    New interpolation spaces and strict Hölder regularity for fractional abstract Cauchy problem
    (01-12-2021)
    Alam, Md Mansur
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    Baleanu, Dumitru
    We know that interpolation spaces in terms of analytic semigroup have a significant role into the study of strict Hölder regularity of solutions of classical abstract Cauchy problem (ACP). In this paper, we first construct interpolation spaces in terms of solution operators in fractional calculus and characterize these spaces. Then we establish strict Hölder regularity of mild solutions of fractional order ACP.
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    Publication
    On Fractional Semilinear Nonlocal Initial Value Problem with State Dependent Delay
    (01-01-2022)
    Alam, Md Mansur
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    In this paper, we consider a class of fractional order semilinear abstract Cauchy problem with state dependent delay subject to nonlocal initial conditions, and enlarge the existence theory with two different sets of assumptions. Under the first set of assumptions, we establish the existence of Hölder classical solution. Since the Hölder exponent appears as an exponent on the metric function in contraction inequality, it is not suitable to use Banach contraction mapping principle. Krasnoselskii’s fixed point theorem becomes effective to overcome this situation. Under the second set of assumptions, we obtain only the existence of mild solution using Schauder’s fixed point theorem. Few examples have been provided to illustrate our results.