Let $(-A,B,C)$ be a linear system in continuous time $t \gt 0$ with input and output space ${\mathbb C}$ and state space $H$. The scattering (or impulse response) functions $\phi_{(x)}(t)=Ce^{-(t+2x)A}B$ determines a Hankel integral operator $\Gamma_{\phi_{(x)}}$; if $\Gamma_{\phi_{(x)}}$ is trace class, then the Fredholm determinant $\tau (x)=\det (I+\Gamma_{\phi_{(x)}})$ determines the tau function of $(-A,B,C)$. The paper establishes properties of algebras including $R_x = \int_x^\infty e^{-tA}BCe^{-tA}\,dt$ on $H$, and obtains solutions of the Kadomtsev–Petviashvili PDE. Pöppe’s semi-additive operators are identified with orbits of a shift action on integral kernels, and Pöppe’s bracket operation is expressed in terms of the Fedosov product. The paper shows that the Fredholm determinant $\det (I+R_x)$ gives an effective method for numerical computation of solutions of $KP$.

The present analysis has been developed to investigate the heat transfer phenomenon in peristaltic flow of Carreau fluid in a curved channel with rhythmic contraction and expansion of waves along the walls (similar to blood flow in tubes). Magnetic field is imposed in radial direction. The heat transfer aspect is further studied with viscous dissipation effect. The curved channel walls are influenced by flow and thermal partial slip. In addition the flow stream comprised porous medium. The system of relevant non-linear PDEs have been reduced to ODEs by utilizing the long wavelength approximation. The striking features of flow and temperature characteristics under the involved parameters are examined by plotting graphs. The generation of fluid temperature and velocity due to viscous dissipation and gravitational efforts are recorded respectively. Moreover indicated results signify activation of velocity, temperature and heat transfer rate with Darcy number.

In order to solve general kinematics modeling problems and numerical stability problems of numerical methods for spatial parallel linkage mechanisms, a general modeling method and its numerical solving algorithm is proposed. According to the need for avoiding direct singular configurations, valid joint variable space and valid forward kinematics solutions (VKSs) are defined. Taking numerical convergence near singular points into account, the pseudo-arc length homotopy continuation algorithm is given to solve the kinematics model. Finally as an example, the joint variable space of the general Stewart platform mechanism is analyzed, which is proved to be divided into subspaces by direct singular surfaces. And then, forward kinematics solutions of 200 testing points are solved separately using the pseudo-arc length homotopy continuation algorithm, the Newton homotopy continuation algorithm and the Newton–Raphson algorithm (NRA). Comparison of the results shows that the proposed method is convergent to the same solution branch with the initial configuration on all the testing points, while the other two algorithms skip to other solution branches on some near singular testing points.

In this study, Liénard equations in their general form are treated using the Adomian decomposition method. The special structure of the Liénard equation is exploited to obtain a numerically efficient algorithm suitable for solution by a computer program.

We investigate the economic importance of modeling nonlinearities in the dynamics of exogenous processes on the implied moments of endogenous variables in the context of the consumption-based asset pricing model. For this purpose, we model the endowment process alternatively as a linear autoregression and as a nonlinear threshold autoregression. The asset pricing model with nonlinear endowment is solved using quadrature techniques. A comparison of the moments of the model-implied rates of return in the two cases suggests that the economic impact of modeling nonlinearities is small.