By using the geometric constraints on the control polygon of a Pythagorean hodograph (PH) quartic curve, we propose a sufficient condition for this curve to have monotone curvature and provide the detailed proof. Based on the results, we discuss the construction of spiral PH quartic curves between two given points and formulate the transition curve of a $G^2$ contact between two circles with one circle inside another circle. In particular, we deduce an attainable range of the distance between the centers of the two circles and summarize the algorithm for implementation. Compared with the construction of a PH quintic curve, the complexity of the solution of the equation for obtaining the transition curves is reduced.

In this paper, the approximate solutions for two different type of two-dimensional
nonlinear integral equations: two-dimensional nonlinear Volterra-Fredholm integral equations
and the nonlinear mixed Volterra-Fredholm integral equations are obtained using the Laguerre
wavelet method. To do this, these two-dimensional nonlinear integral equations are transformed
into a system of nonlinear algebraic equations in matrix form. By solving these systems, unknown coefficients are obtained. Also, some theorems are proved for convergence analysis.
Some numerical examples are presented and results are compared with the analytical solution
to demonstrate the validity and applicability of the proposed method.