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Particle scattering
Published in Timothy R. Groves, Charged Particle Optics Theory, 2017
We have succeeded in reducing the dimensionality to a single integration variable η. This technique in the theory of partial differential equations is known as the method of characteristics. We can now proceed to perform the integration as lnF˜(k,l,z)=12kµ∫ξdη[1−τ˜(l)], where the subscript ξ signifies that ξ must be kept constant over the integration path. We treat the variable k as constant, as no derivative of k appears. We also make use of 2l=ξ+η,2dl=dξ+dη.
Analytical methods
Published in Charles Aubeny, Geomechanics of Marine Anchors, 2017
The method of characteristics is a technique for solving hyperbolic partial differential equations by transforming them into a family of ordinary differential equations that can be solved by integration along so-called characteristic surfaces (Rao, 2002). In bearing capacity problems, the equations of equilibrium coupled with the yield criterion generate a system of hyperbolic differential equations amenable to solution by this method. As this solution approach enforces static equilibrium everywhere, but does not ensure strain compatibility, computed collapse loads should be considered as lower bounds. The example presented below is for a plane strain problem, but solutions to axisymmetric problems are also possible.
The hydraulic impact and alleviation phenomena numeric modeling in the industrial pumped pipelines
Published in Genadiy Pivnyak, Oleksandr Beshta, Mykhaylo Alekseyev, Power Engineering, Control and Information Technologies in Geotechnical Systems, 2015
To solve this problem in this paper the method of characteristics (Bakar & Firoz 2002.) has been used. The method of characteristics converts partial differential equations, for which the solution can’t be written in general terms (as, for example, the equations describing the fluid flow in a pipe) into the equations in total derivatives. The resulting nonlinear equations can then be integrated using the methods of using the equations of finite differences.
Central arterial pressure and patient-specific model parameter estimation based on radial pressure measurements
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2023
Dániel Gyürki, Tamás Horváth, Sára Till, Attila Egri, Csilla Celeng, György Paál, Béla Merkely, Pál Maurovich-Horvat, Gábor Halász
The governing equation system is discretized and solved using the method of characteristics. This method transforms the governing partial differential equations (PDE) to ordinary differential equations by finding the characteristic curves or characteristics of the PDE, on which a certain quantity of the system is constant. The original quantities, like pressure or velocity can be calculated at the intersection of these characteristics. Originally, the method of characteristics marches forward in time, when the pressures are known at both ends of the system. However, Bárdossy and Halász (2013) proposed a method, in which the upstream boundary (e.g. the aortic pressure curve) can be calculated based on a peripheral pressure measurement.
Central arterial pressure estimation based on two peripheral pressure measurements using one-dimensional blood flow simulation
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2023
Dániel Gyürki, Péter Sótonyi, György Paál
The equation system describing the flow in viscoelastic tubes consists of the modified continuity equation, the momentum equation and the equations of the Poynting-Thomson (or Stuart) model responsible for the viscoelasticity of the arterial walls. The method of characteristics transforms these equations into ordinary differential equations by finding the characteristic curves of the partial differential equations (PDE). The characteristic curves march forward in time in the original MoC method, but Bárdossy and Halász (2013) described the method of characteristics marching backwards in space segmentwise.