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A Priori Estimates for Solutions of Boundary Value Problems for Fractional-Order Equations
A. A. Alikhanov
TL;DR
Boundary value problems for the diffusionwave equation require a priori estimates for their solutions. The paper uses energy inequalities and related integral estimates to establish such bounds for first- and third-kind problems under stated coefficient conditions.
Problem
The paper addresses how to obtain a priori estimates for solutions of first- and third-kind boundary value problems for the diffusionwave equation.
Method
The paper applies energy inequalities and the Gronwall-Bellman lemma to derive estimates for the considered boundary value problems.
Results
The paper establishes a priori estimates for first- and third-kind problems, with one estimate implying existence and continuous dependence on input data.
Takeaways & Limitations
Under the stated regularity, positivity, and boundedness assumptions, the estimates provide solution bounds and, for one problem, existence and continuous dependence on the input data.
Takeaways & Limitations
The analysis assumes that a sufficiently regular solution already exists for the boundary value problems considered.
Abstract
from arXiv · showhide
We consider boundary value problems of the first and third kind for the diffusionwave equation. By using the method of energy inequalities, we find a priori estimates for the solutions of these boundary value problems.
1 BOUNDARY VALUE PROBLEMS FOR THE FRACTIONAL DIFFUSION EQUATION
The section derives a priori estimates for first and third boundary value problems for the fractional diffusion equation under specified coefficient, boundary-data, and regularity assumptions. These estimates imply uniqueness and continuous dependence for the first problem.
- First boundary value problem: Under k(x, t) ∈ C1,0( ¯QT), q(x, t) ≥ 0, and k(x, t) ≥ c1 > 0, Theorem 1 establishes an a priori estimate for the first boundary value problem.The solution is assumed to belong to C2,1( ¯QT).
- First boundary value problem: The first problem’s a priori estimate yields uniqueness and continuous dependence of the solution on the input data.
- Third boundary value problem: The third-problem estimate is obtained using a generalized Gronwall-Bellman lemma and properties of Mittag-Leffler functions.The argument applies the lemma to a fractional differential inequality for the spatial norm of the solution.
- Third boundary value problem: The third problem’s estimate incorporates the forcing term, boundary data µ1(t) and µ2(t), and initial value u0(x).
2 BOUNDARY VALUE PROBLEMS FOR THE FRACTIONAL WAVE EQUATION
The section derives a priori estimates for first and third boundary value problems for the fractional wave equation under regularity, positivity, and boundary-coefficient assumptions. These estimates imply existence and continuous dependence on input data for the first problem.
- First boundary value problem: The first problem imposes homogeneous endpoint conditions u(0,t) = 0 and u(l,t) = 0 together with initial data u(x,0) = u0(x) and ut(x,0) = u1(x).The equation contains the Caputo fractional derivative of order 1 + α, with 0 < α < 1.
- First boundary value problem: The a priori estimate for the first problem implies that its solution exists and continuously depends on the input data.The paper assumes a solution u(x,t) ∈ C2,2(¯Q) while deriving the estimate.
- First boundary value problem: Theorem 3 establishes an a priori estimate for the first boundary value problem when k, q, and f satisfy stated regularity and boundedness conditions.The assumptions include k ∈ C1,1(¯QT), q ∈ C0,1(¯QT), f ∈ C(¯QT), 0 < c1 ≤ k ≤ c2, 0 < m1 ≤ q ≤ m2, and bounded time derivatives of k and q.
- Energy-estimate method: Both estimates are obtained through energy identities and inequalities, followed by applications of the Gronwall-Bellman lemma and, for the third problem, Lemma 2.The third-problem proof uses boundary-term inequalities involving βi and µi before applying these tools.
- Third boundary value problem: Theorem 4 establishes an a priori estimate for the third boundary value problem under Theorem 3’s assumptions plus regularity and positivity conditions on βi and µi.The additional conditions are βi(t), µi(t) ∈ C1[0,T], βi(t) ≥ β > 0, and |βit(t)| ≤ c4 for i = 1, 2.