Considering economic variables changing from time to time, the time-varying models can fit the financial data better. In this paper, we construct stochastic volatility models with time-varying coefficients. Furthermore, the interest rate risk is one of important factors for timer options pricing. Therefore, we study the timer options pricing for stochastic volatility models with changing coefficients under time-varying interest rate. Firstly, the partial differential equation boundary value problem is given by using \(\Delta\)-hedging approach and replicating a timer option. Secondly, we obtain the joint distribution of the variance process and the random maturity under the risk neutral probability measure. Thirdly, the explicit formula of timer option pricing is proposed which can be applied to the financial market directly. Finally, numerical analysis is conducted to show the performance of timer option pricing proposed.

A monotone iterations algorithm combined with the finite difference method is constructed for an obstacle problem with semilinear elliptic partial differential equations of second order. By means of Dirac delta function to improve the computation procedure of the discretization, the finite difference method is still practicable even though the obstacle boundary is irregular. The numerical simulations show that our proposed methods are feasible and effective for the nonlinear obstacle problem.

This paper is concerned with the existence and non-existence of traveling wave solutions for a diffusive SIR model with delay and nonlinear incidence. First, we construct a pair of upper and lower solutions and a bounded cone. Then we prove the existence of traveling wave by using Schauder's fixed point theorem and constructing a suitable Lyapunov functional. The nonexistence of traveling wave is obtained by two-sided Laplace transform. Moreover, numerical simulations support the theoretical results. Finally, we also obtain that the minimal wave speed is decreasing with respect to the latent period and increasing with respect to the diffusion rate of infected individuals.

This article presents an approximate numerical solution for nonlinear Duffing Oscillators by pseudospectral (PS) method to compare boundary conditions on the interval [-1, 1]. In the PS method, we have been used differentiation matrix for Chebyshev points to calculate numerical results for nonlinear Duffing Oscillators. The results of the comparison show that this solution had the high degree of accuracy and very small errors. The software used for the calculations in this study was Mathematica V.10.4.

In this paper, we investigate the large-time behavior of a nonlinear size-structured population model with logistic term and \(T\)-periodic vital rates. We establish the existence of a unique non-negative solution of the given model with the given initial distribution. We prove that there exists at most two \(T\)-periodic non-negative solutions (one of them being the trivial one) of the periodic model associated with the given model. We show that for any initial distribution of population the solution of the given model tends to the nontrivial non-negative \(T\)-periodic solution of the associated model. At last, we give the numerical tests, which are used to demonstrate the effectiveness of the theoretical results in our paper.

The paper reports on an iteration algorithm to compute asymptotic solutions at any order for a wide class of nonlinear singularly perturbed difference equations.

The aim of this article is to investigate the global existence and blow-up behavior of the nonnegative solution to a degenerate and singular parabolic equation with nonlocal boundary condition. The conditions on the existence and non-existence of the global solution are given. Furthermore, under some appropriate hypotheses, the precise blow-up rate estimate and the uniform blow-up profile of the blow-up solutions are discussed.

Let \(K\) be a nonempty closed convex subset of a Banach space \(E\) and \(T: K\rightarrow K\) be a nonexpansive mapping. Using a viscosity approximation method, we study the implicit midpoint rule of a nonexpansive mapping \(T.\) We establish a strong convergence theorem for an iterative algorithm in the framework of uniformly smooth Banach spaces and apply our result to obtain the solutions of an accretive mapping and a variational inequality problem. The numerical example which compares the rates of convergence shows that the iterative algorithm is the most efficient. Our result is unique and the method of proof is of independent interest.