In this paper, we study existence of solutions of mean field equations for the equilibrium turbulence and Toda systems on connected finite graphs. Our method is based on calculus of variations, which was built on connected finite graphs by Grigor'yan, Lin and Yang.

In this paper, we obtained the extremal function for a weighted singular Trudinger-Moser inequality by  blow-up analysis in the Euclidean space $\mathbb{R}^2$. This extends recent results of Hou (J. Inequal. Appl.,  2018) and similar result was proved by Zhu (Sci. China Math., 2021).

In this paper, an averaging principle for the solutions to mixed stochastic differential equation involving standard Brownian motion, a fractional Brownian motion $B^{H}$ with the Hurst parameter $H>\frac{1}{2}$ and a discontinuous drift was estimated. Under some proper assumptions, we proved that the solutions of the simplified systems can be approximated to that of the original systems in the sense of mean square by the method of the pathwise approach and the It$\hat{\mathrm{o}}$ stochastic calculus.

In the paper, the asymptotic behavior of the solution for the parabolic equation system of porous media coupled by three variables and with weighted nonlocal boundaries and nonlinear internal sources is studied. by constructing the upper and lower solutions with the ordinary differential equation as well as introducing the comparison theorem, the global existence and finite time blow-up of the solution of parabolic equations of porous media coupled by the power function and the logarithm function are obtained. The differential inequality technique is used to obtain the lower bounds on the blow up time of the above equations under Dirichlet and Neumann boundary conditions.

This paper, we study the multiplicity of solutions for the fractional Schr${\rm\ddot{o}}$dinger equation

\begin{equation*}(-\Delta)^su+V(x)u=u^p,\ \ u>0,\ \ x\in\mathbb{R}^N,\ \ u\in H^s(\mathbb{R}^N),\end{equation*}

with $s\in(0,1),\  N\geq3,\  p\in(1,\frac{2N}{N-2s}-1)$ and $\lim_{|y|\rightarrow+\infty}V(y)>0$. By assuming suitable decay property of the radial potential $V(y)=V(|y|)$, we construct another type of solutions concentrating at infinite vertices of two similar equilateral polygonal with infinitely large length of sides. Hence, besides the length of each polygonal, we must consider one more parameter, that is the height of the podetium,  simultaneously. Another difficulty lies in the non-local property of the operator $(-\Delta)^s$ and the algebraic decay involving the approximation solutions make the estimates  become more subtle.

In this paper we have discussed solution and stability analysis of ordinary and partial differential equation with boundary value problem. We investigated periodic stability in Euler’s scheme and also discussed PDE’s by finite difference scheme. Numerical example has been discussed finding nature of stability.  All given result more accurate other than existing methods.

In this paper, we consider the two-dimensional (2D) Prandtl-Hartmann equations on the half plane and prove the  global existence and uniqueness of solutions to 2D Prandtl-Hartmann  equations by using the classical energy methods in analytic framework. We prove that the lifespan of the solutions to 2D Prandtl-Hartmann  equations can be extended up to $T_\varepsilon$ (see Theorem 2.1) when  the strength of the perturbation is of the order of $\varepsilon$. The difficulty of solving the Prandtl-Hartmann equations in the analytic framework is the loss of $x$-derivative in the term $v\partial_yu$. To overcome this difficulty, we introduce the Gaussian weighted Poincar$\acute{e}$ inequality (see Lemma 2.3). Compared to the existence and uniqueness of solutions to the classical Prandtl equations where the monotonicity condition of the tangential velocity plays a key role, which is not needed for the 2D Prandtl-Hartmann  equations  in analytic framework. Besides, the existence and uniqueness of solutions to the 2D MHD boundary layer where the initial tangential magnetic field has a lower bound plays an important role, which is not needed for the 2D Prandtl-Hartmann  equations  in analytic framework, either.