Deterministic Source Terms Research Articles

Inverse problems for the fractional diffusion equation driven by fractional Brownian sheet

Abstract This paper explores the inverse problems of the space-time fractional diffusion equation driven by a space-time fractional Brownian sheet, including spatial stochastic source term, spatial deterministic source term, potential, and fractional order. In order to deal with the stochastic integral of fractional Brownian sheet with space-time Hurst indexes $\mathcal{H}_1 , \mathcal{H}_2 \in (0,1)$, we construct a unified Itô isometry framework for the first time, upon which we establish the regularity of the solution to the direct problem. Then, we ingeniously employ the generalized fractional Grönwall inequality, Sobolev embedding, and combine the statistical characteristics of the outward normal derivative of the weak solution on open subdomain at the boundary to achieve accurate recovery of the stochastic space-time source terms and prove the uniqueness of the space deterministic source term. Furthermore, by constructing a contraction mapping and exploiting the Banach fixed point theorem, we establish the uniqueness of the potential. Afterward, using the statistical characteristics of the solution at the terminal time, we establish the stability of the potential. Finally, leveraging the representation of the mild solution and the asymptotic properties of the Mittag-Leffler function, we examine the uniqueness and stability of fractional order at known terminal time. In scenario involving an unknown terminal time, we derive the uniqueness and stability of the terminal time.

Neural Network Modeling of Deterministic Unsteadiness Source Terms

A neural-network-based lumped deterministic source term technique is presented that results in the prediction of an approximate time-average solution when used to modify a steady-state solver. Three different neural networks are developed for simple cavity flows using Mach number, cavity length-to-depth ratio, and aft wall translation as parameters. The results indicate that axial force data can be reproduced with less than 15% error as compared to the time average of a fully unsteady calculation. Computation times for the resultant neural network lumped deterministic source term approach were up to two orders of magnitude less than the comparable unsteady solution and were essentially identical to that of a steady-state calculation, although it must be noted that a database of unsteady calculations is required to develop the technique. The lumped deterministic source terms did not appear to affect the robustness of the steady-state solver adversely.

Linear Deterministic Source Terms for Hot Streak Simulations

Steady state surface rothalpy over specific heat at constant pressure (temperature units) results obtained with a linear unsteady solution-based lumped deterministic source term are compared with results obtained from a traditional, nonlinear, inviscid unsteady solution for an aircraft engine first stage high-pressure turbine rotor configuration. The relationship between the source terms and traditional solution variables is explored to offer a unique insight into comparing the two approaches. Boundary condition/potential field effects and the order of accuracy of the available schemes are also explored and show a significant effect on surface rothalpy results. The new technique demonstrates a significant potential for approximately including unsteady hot streak effects in time average calculations with minimal computer effort.