[Paper Review] Non-local boundary value problem for a mixed-type equation involving the bi-ordinal Hilfer fractional differential operators

Erkinjon Karimov, Bakhodirjon Toshtemirov|arXiv (Cornell University)|Jun 24, 2021

Differential Equations and Boundary Problems16 references4 citations

TL;DR

This paper investigates the existence and uniqueness of solutions for a non-local boundary value problem involving a mixed-type partial differential equation with bi-ordinal Hilfer fractional derivatives in a rectangular domain. By employing eigenfunction expansion and reduction to Volterra integral equations, the authors establish sufficient conditions on the data ensuring solution existence and uniform convergence of the series solution via the Weierstrass M-test.

ABSTRACT

In this paper, we consider a nonlocal boundary-value problem for a mixed-type equation involving the bi-ordinal Hilfer fractional derivative in a rectangular domain. The main target of this work is to analyze the uniqueness and the existence of the solution of the considered problem by means of eigenfunctions. Moreover, we construct the solution of the ordinary fractional differential equation with the right-sided bi-ordinal Hilfer derivative by the method of reduction to the Volterra integral equation. Then, we present sufficient conditions for given data in order to show the existence of the solution.

Motivation & Objective

• To analyze the existence and uniqueness of solutions for a non-local boundary value problem involving mixed-type equations with bi-ordinal Hilfer fractional derivatives.
• To extend the theory of fractional differential equations by incorporating the bi-ordinal Hilfer operator, which generalizes Riemann-Liouville and Caputo derivatives.
• To establish sufficient conditions on the given data (source function, initial, and non-local conditions) ensuring the solution's regularity and convergence.
• To demonstrate the convergence of the series representation of the solution and its derivatives using the Weierstrass M-test.

Proposed method

• The problem is formulated in a mixed domain Ω = Ω₁ ∪ Ω₂ ∪ AB, with parabolic and hyperbolic parts governed by left- and right-sided bi-ordinal Hilfer fractional derivatives.
• The solution is constructed via eigenfunction expansion in the spatial variable, reducing the PDE to a system of ordinary fractional differential equations.
• The ordinary fractional differential equations with right-sided bi-ordinal Hilfer derivatives are reduced to Volterra integral equations of the second kind.
• The convergence of the series solution is proven using the Weierstrass M-test, relying on bounds derived from the Mittag-Leffler functions and L2 norms of derivatives.
• Sufficient conditions on the data are derived by expressing the solution coefficients in terms of Fourier coefficients of ψ(x), f(x,t), and their derivatives.
• The analysis includes conjugation conditions on the interface t=0, ensuring continuity of the fractional integral of the solution across the mixed-type boundary.

Experimental results

Research questions

• RQ1 Under what conditions does a non-local boundary value problem for a mixed-type equation with bi-ordinal Hilfer fractional derivatives admit a unique solution?
• RQ2 How can the solution be represented as a uniformly convergent series in a rectangular domain?
• RQ3 What role do the eigenfunction expansions and Volterra integral equations play in proving existence and uniqueness?
• RQ4 How do the regularity and integrability conditions on the data (ψ(x), f(x,t)) affect the solution's smoothness and convergence?
• RQ5 What are the precise sufficient conditions on the source function and non-local data to ensure the solution's boundedness and continuity?

Key findings

• The solution exists and is unique under the conditions that Δₙ ≠ 0, ψ(x) ∈ C[0,l] ∩ C⁴(0,l) with ψ(0)=ψ(l)=0, ψ''(0)=ψ''(l)=0, ψ⁽⁴⁾(0)=ψ⁽⁴⁾(l)=0, and ψ⁽⁵⁾(x) ∈ L₂(0,l).
• The source function f(x,t) must satisfy f(0,t)=f(l,t)=0, fₓₓ(0,t)=fₓₓ(l,t)=0, f(x,t) ∈ C[0,l]×[-T,T] ∩ C²,¹(0,l)×(-T,T), and fₓ⁽³⁾(·,t) ∈ L₂(0,l).
• The series representation of uₓₓ(x,t) converges uniformly in Ω₁ ∪ Ω₂, as guaranteed by the Weierstrass M-test and bounds on Mittag-Leffler functions.
• The series for the fractional derivatives D₀₊⁽ᵃ¹,ᵇ¹⁾ᵘ¹u and D₀₋⁽ᵃ²,ᵇ²⁾ᵘ²u also converge uniformly, following the same argument as for uₓₓ.
• The coefficients νₙ, φₙ, and τₙ are explicitly expressed in terms of the Fourier coefficients of ψ(x) and f(x,t), with νₙ involving the inverse of Δₙ and L₂ norms of derivatives.
• Parseval’s identity and the inequality 2|1/Δₙ√λₙ ψ₅ₙ| ≤ 1/Δₙ²λₙ + |ψ₅ₙ|² are used to bound the series and ensure convergence.