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Differential Equations of Thermodynamics
Published in Ziya Uddin, Mukesh Kumar Awasthi, Rishi Asthana, Mangey Ram, Computing and Simulation for Engineers, 2022
Normally, the beginners of applied mathematics and mathematical physics learn the fundamental laws from the perspective of a systems approach, for which the inventory of mass remains fixed. Contrarily, many physical problems of practical interest demand a control volume approach, for which the entity of mass varies. For some reasons, it is advisable to reformulate all the pertinent laws of physics from the standpoint of a control volume. This in turn demands a general formulation that can pave the way for a smooth transition from a systems approach to the control volume approach and vice versa. The Reynolds transport theorem (RTT) (Reynolds, 1903; Granger, 2020; Shames, 1992) is such a mathematical apparatus that serves the purpose. The RTT is also known as Leibnitz–Reynolds transport theorem. It is a three-dimensional generalization of Leibniz integral rule or simply differentiation under the integral sign. It is to be remarked here that in the fluid mechanics literature, the systems approach is recognized as material or Lagrangian description and the control volume approach as spatial or Eulerian description although historians argue that both descriptions should be attributed to Euler (Yih, 1979).
Mathematical Modelling in Food Science through the Paradigm of Eggplant Drying
Published in Surajbhan Sevda, Anoop Singh, Mathematical and Statistical Applications in Food Engineering, 2020
Alessandra Adrover, Antonio Brasiello
which, after applying both the Leibniz integral rule and Gauss Theorem, becomes: () ∂∂t∫VϕdV=D∫Ω∇ϕ⋅ndΩ−∫Ωϕ(v−vΩ)⋅ndΩ
Moving load identification on Euler-Bernoulli beams with viscoelastic boundary conditions by Tikhonov regularization
Published in Inverse Problems in Science and Engineering, 2021
Guandong Qiao, Salam Rahmatalla
During the load duration ([0, L/Ve]), the vertical displacement response can be obtained by the convolution integral based on Equation (A24) in the appendix. The vertical acceleration response can be derived using two derivative operations by the Leibniz integral rule. Equation (15) can then be rewritten in discrete terms. where Using the definition from Equation (17) to simplify Equation (16), In this case, it becomes easier to determine the contribution on the acceleration responses from the moving load f(t) and its first derivation . Taking location xl as example, its acceleration response can be given in the matrix form based on Equation (18), when the first derivation of input is approximated by the central difference method [36]. The measured duration of the acceleration response can be longer than or equal to the duration of the moving load (N ≥ Nf). when N > Nf, and .
Adjoint-based sensitivity analysis of steady char burnout
Published in Combustion Theory and Modelling, 2021
Ahmed Hassan, Taraneh Sayadi, Vincent Le Chenadec, Heinz Pitsch, Antonio Attili
The two problems are related by the Leibniz integral rule: or stated otherwise, the gradient of g matches the sensitivity of the gradient of with respect to s. As a result, both gradients can be computed by the adjoint method as follows: Solution of the primal two-point boundary value problem, by means of shooting or collocation methods, as highlighted above.Solution of the adjoint of the primal problem. This is a linear multi-point boundary value problem, which, in the limits and degenerates into a linear two-point boundary value problem.
Generalized Hybrid Censored Reliability Acceptance Sampling Plans for the Weibull Distribution
Published in American Journal of Mathematical and Management Sciences, 2018
Tanmay Sen, Ritwik Bhattacharya, Yogesh Mani Tripathi, Biswabrata Pradhan
The expression of E(ξ) is given by (See Park & Balakrishnan, 2009) , where and are the expected duration of the experiments under Type-I HCSs (n, r, Xl) and (n, l, Xl), respectively. They are given by, and Now, by using (14) and (15), E(ξ) can be written as follows. Note, that the cumulative distribution function of Xr: n − 1 is given by, So, for fixed n, r, and l, E(ξ) is a function of Xl only. Then, by using Leibniz integral rule, we obtain, Hence, E(ξ) is a monotone increasing function in Xl.