Orthonormal shifted discrete Legendre polynomials for the variable-order fractional extended Fisher–Kolmogorov equation

The Existence of Solution to Fractional Boundary Value Problem with Riemann-Liouville Type History-State-Based Variable-Order Derivative

“…In this section we devote to dealing with the existence and uniqueness of solution to the boundary value problem (10). We assume that (A 2 ) h ∈ L(0, T);…”

“…Let H η [0, T ](0 < η < 1) be the space of functions x(t) such that x(t) is Hölder continuous with index η on the interval [0, T ] with respect to the norm 28]). For 0 < γ < η < 1, we have (10) (11…”

“…together with x(0) = x(T ) = 0, which implies x ∈ H β [0, T ] is a solution of boundary value problem (10).…”

“…Subsequently, Lorenzo et al [3] and Coimbra [4] studied VO calculus by discussing its possible applications in mechanics, which marked the starting point for applications of VO operators to the analysis of different complex physical problems. After that VO fractional differential equations are widely employed in mechanics and dynamics, viscoelasticity, the modelling of transport processes, control theory because VO fractional operators can describe accurately the memory and hereditary properties of many physical phenomena and processes depending on their nonstationary power-law kernel [5][6][7][8][9][10].…”

“…The paper is organized as follows. In section 2, we present some basic notations, propositions and lemmas, which will play an important role in obtaining the existence of solution to the boundary value problem (10). In section 3, main results are stated based on our previous analysis, then two examples are given to substantiate the theoretical results.…”