Inversion based on differential methods

This expansion is described by twelve constant coefficients for each element 

Omitting the long derivation, we can write the resulting system of algebraic equations for the coefficients of the EE scheme as follows  

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Equations (8) form an infinite system of coupled non-linear partial differential equations for the fransformed potentials,, . For computation purposes, system (8) is also truncated at the Ntii row and colimm, with N sufficiently large for the required convergence. A few automatic numerical integrators for tiiis class of one-dimensional partial differential systems are now readily available, such as those based on tiie Method of Lines [41, 52]. Once the transformed potentials have been computed from numerical solution of system (8), tiie inversion formula Eq.(7.b) is recalled to reconstruct the original potentials, in explicit form along thejc v -iables. 

Kissinger relationship, the most extensively used method in kinetic studies since 1957 [13], was in use to determine the energy of activation and the order of reaction, from plots of the logarithms of the heating rate against the temperature inverse at the maximum reaction rate in isothermal conditions. This method is usually based on the Differential Scanning Calorimetry (DSC) analysis of formation or decomposition processes and in relation to these processes the endothermic and exothermic peak positions are related to the heating rate. 

A method was proposed for the parameterization of impedance based models in the time domain, by deriving the corresponding time domain model equation with inverse Laplace transform of the frequency domain model equation assuming a current step excitation. This excitation signal has been chosen, since it can be easily applied to a Li-ion cell in an experiment, allows the analytical calculation of the time domain model equation and is included in the definition of the inner resistance. The voltage step responses of model elements were presented for lumped elements and derived for distributed model elements that have underlying fractional differential equations using fractional calculus. The determination of the inner resistance from an impedance spectrum was proposed as a possible application for this method. Tests on measurement data showed that this method works well for temperatures around room temperature and current excitation amplitudes up to 10 C. This technique can be used for comparisons of measured impedance spectra with conventionally determined inner resistances. 

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