Sentence examples for using the general solutions from inspiring English sources

Step 4. Solving the equations system in Step 3, and using the general solutions of (1), we can construct a variety of exact solutions for (7).

Using the general solution, analytical expressions of thermal distortion and associated internal forces for commonly used end supports are obtained.

Finally, an application to forming process design problems is discussed and a simple example is illustrated using the general solution.

In this paper, based on the anomalous diffusion phenomena that also have the long-tail behavior, we developed a new fractional decline curve (FDC) model with three fitting parameters using the general solution of the fractional diffusion equations, which is a special case of so-called Mittag-Leffler function.

For free vibration, an analytic form of frequency equation is obtained by using the general solution expressed in terms of some initial parameters at one end of the arch; while for forced vibration, the system's response is obtained analytically by solving a set of algebraic equations with only three unknowns.

The remaining solutions can be obtained using the general solution of the differential equation (3.3) and then imposing the initial condition (3.4).

In the following discussion we will use the general solution (3.25) and its fractional Caputo derivative ^{mathrm{C}}_{0} D^{beta} y(t) = A t^{-beta} E_{alpha, 1-beta}bigl -lambda t^{alpha}bigr) + B t^{1-beta}bigl -lambda-beta }bigl(-lambda t^{alpha}bigr).

According to the physical model, the transmission and reflection matrices (TRM) method was used, and the general solutions for the stresses and displacements of the layered system subjected to dynamic double-circular loads at its surface were obtained.

It is difficult to obtain an a priori estimate of the approximate solution for the study of the existence of global solutions by using the general Galerkin approximation method, but the theory of the potential well often makes up for the defect.

The spatial solutions to the equations of motion are carried out using the general Galerkin method.

The extension of Abel's Identity presented in the companion paper is used to construct the general solutions to some non-linear, autonomous systems.