Generalized Bessel QuasilinearizationTechnique Applied to Bratu and Lane–Emden-Type Equations of Arbitrary Order

The ultimate goal of this study is to develop a numerically effective approximation technique to acquire numerical solutions of the integer and fractional-order Bratu and the singular Lane–Emden-type problems especially with exponential nonlinearity. Both the initial and boundary conditions were considered and the fractional derivative being considered in the Liouville–Caputo sense. In the direct approach, the generalized Bessel matrix method based on collocation points was utilized to convert the model problems into a nonlinear fundamental matrix equation. Then, the technique of quasilinearization was employed to tackle the nonlinearity that arose in our considered model problems. Consequently, the quasilinearization method was utilized to transform the original nonlinear problems into a sequence of linear equations, while the generalized Bessel collocation scheme was employed to solve the resulting linear equations iteratively. In particular, to convert the Neumann initial or boundary condition into a matrix form, a fast algorithm for computing the derivative of the basis functions is presented. The error analysis of the quasilinear approach is also discussed. The effectiveness of the present linearized approach is illustrated through several simulations with some test examples. Comparisons with existing well-known schemes revealed that the presented technique is an easy-to-implement method while being very effective and convenient for the nonlinear Bratu and Lane–Emden equations.

Stability and prevalence of McKean–Vlasov stochastic differential equations with non-Lipschitz coefficients

We consider various approximation properties for systems driven by a McKean–Vlasov stochastic differential equations (MVSDEs) with continuous coefficients, for which pathwise uniqueness holds. We prove that the solution of such equations is stable with respect to small perturbation of initial conditions, parameters and driving processes. Moreover, the unique strong solutions may be constructed by an effective approximation procedure. Finally, we show that the set of bounded uniformly continuous coefficients for which the corresponding MVSDE have a unique strong solution is a set of second category in the sense of Baire.

Approximation of a function describing a quality criterion of the thermo-mechanical fusible interfacing process

The present work aims to investigate the function describing the relationship between a quality criterion and input factors of the thermo-mechanical fusible /TMF/ interfacing process and to derive its effective approximation. An approximation by interpolation was applied for the purpose of the study. A numerical realization of a linear and exponential approximation of the mathematical model describing the TMF interfacing process was performed. An effective linear approximation of the function connecting the quality criterion with the input factors of the TMF interfacing process was found. This creates conditions for replacing the relatively complex function (describing the TMF interfacing process) with its linear approximation. The linear approximation gives the possibility easier and faster to determine the relationships between the input factors and the quality criterion. This created conditions for ignoring the subjective factor and for optimizing and automating the studied technological process.