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Computational Numerical Methods
Published in Timothy Bower, ®, 2023
Equation (7.10) is an implicit equation for solving systems of ODEs numerically, and is implemented in the IBEsysODE function listed in code 7.16. Notice that the implicit approach requires solving an equation at every step. In this case, the equation is a linear system of equations with a fixed matrix, so the IBEsysODE function uses LU decomposition once and calls on Matlab's triangular solver via the left-divide operator to quickly find a solution in each loop iteration. If this were a nonlinear problem, then numerical methods such as Newton's method or the secant method described in section 7.1.1 would be needed, which could add significant computation. For this reason, implicit methods are not recommended for problems that are either not stiff or nonlinear [65].
Theory of Type II Superconductors
Published in R. D. Parks, Superconductivity, 2018
Alexander L. Fetter, Pierre C. Hohenberg
Equations (45), (46), and (75) yield an implicit equation for the constitutive relation B = Be (H) : () H=Hc1+(ϕ0/8πλ2)∑j′{2K0(rj/λ)K1(rj/λ)}
Differential Equation Problems
Published in Dingyü Xue, YangQuan Chen, Scientific Computing with MATLAB®, 2018
The implicit difierential equation solver ode15i() can also be used to solve this problem. An anonymous function can be written to describe the implicit equation. Let x0 = [0.8,0.1,*]’ andx0F=[1,1,0]', where * is used to denote free values. The following statements can be used to find compatible initial conditions. The original difierential algebraic equation can be solved in this way, and the same results can be obtained. It can be seen that with the use of the method, a more straightforward solution process can be enjoyed.
Coupling of approximate convection diffusion wave method with ultrasonic sensor to estimate discharge in Himalayan Rivers
Published in ISH Journal of Hydraulic Engineering, 2023
Kirtan Adhikari, Chokey Yoezer, Jit Bahadur Mongar, Sangpo Tamang, Tenzin Phuentsh, Tshering Tashi
Channel roughness is one such prominent factor that predominantly affects discharge estimation. On a technical note, the roughness is responsible for frictional losses within a river reach and depends upon the surface roughness of the material over which it flows and the flow regime, usually expressed in Reynolds number. The influence of frictional factors is extensively studied for pipe flows. For example, Colebrook – White equation is an implicit equation expressed as: