As an important problem in the field of information science such as machine learning and image processing, low rank matrix completion has been widely studied. The first-order primal-dual algorithm is one of the classical algorithms for solving this problem. However, the data processed in practical applications is often large-scale. Therefore, based on the framework of the primal-dual algorithm, this paper proposes a modified primal-dual algorithm for large-scale matrix completion problems by exploring correction strategy with the variable step size. In each iteration of the new algorithm, the primal and dual variables are firstly updated by the primal-dual algorithm, and then the correction strategy with the variable step size is used to further correct the two variables. Under certain assumptions, the global convergence of the new algorithm is proved. Finally, the new algorithm is verified to be efficient by solving some random low rank matrix completion problems and examples of image restoration.

In view of the practical problems of malaria vaccine in malaria transmission, a dynamical model of malaria with vaccination and vaccine failure is constructed, and the control reproduction number $\mathcal{R}_{c}$ is calculated. The existence conditions of malaria-free and malaria equilibria in terms of $\mathcal{R}_{c}$ are given. By using the Lyapunov function method and the generalized Lyapunov-Lasalle theorem, the sufficient and necessary conditions for the global asymptotic stability of malaria-free and malaria equilibria with respect to $\mathcal{R}_{c}$ are established.

BAR (broker adaptive ridge) is a new method to replace $L_0$ penalty regression. Based on a reweighted $L_2$ penalty, BAR penalty combines the advantages of two different penalties, $L_0$ and $L_2$, and avoids the shortcomings of separately using these two penalties. In this paper, the BAR method is extended to linear regression model with robust loss function, and the coordinate descent algorithm is used to estimate the parameters. To characterize the robustness of the proposed method, we provide an influence function for robust BAR estimation. Under appropriate conditions, we theoretically establish the variable selection consistency and Oracle property of BAR estimation, and provide proof. We compared the proposed method with other existing methods through numerical simulation and analysis of real data, further verifying that the new method is more effective in terms of robustness and variable selection performance.

In this paper, the partial functional linear multiplicative model is considered. This model, which becomes partial functional linear regression after taking logarithmic transformation, is useful in analyzing data with positive responses. Based on B-splines, two estimation methods are proposed by minimizing the least absolute relative error (LARE) and the least product relative error (LPRE), respectively. The dimension of the B-spline bases is selected using the Schwarz information criteria. Consistency and asymptotic normality of the two methods are investigated. For the slope function, we prove that its convergence rate achieves the optimal rate of nonparametric function. Monte Carlo simulations are conducted to evaluate and compare the finite sample performance of the proposed estimators with the least squares (LS) estimator and the least absolute deviation (LAD) estimator under the different random error settings. Simulation results show that the proposed methods are comparable to other methods. Finally, an example of real data analysis is given to illustrate the application of the model.

The modified phase field crystal model is a sixth order nonlinear generalized damped wave equation. Based on the Crank-Nicolson scheme, a second order implicit linearized finite difference scheme is presented by using the method of order reduction. The nonlinear term is approximated by the second order explicit extrapolation. A theoretical analysis is carried out by the energy argument and mathematical induction. The unique solvability and $L^{\infty}$ norm convergence of the numerical scheme are proved rigorously. The convergence order is two in time and space. Numerical results demonstrate that the presented scheme for the modified phase field crystal equation is efficient and can achieve the expected accuracy.

As a generalization of integer-order differential equations, fractional differential equations have been widely used in science and engineering in recent years, which also have attracted widespread attention from many scholars. In this paper, a novel Crank-Nicolson finite volume method (CN-FVM) is proposed for solving Riesz space-fractional advection-diffusion equations (RSFADEs) with homogeneous Dirichlet boundary conditions. In order to obtain the discrete linear systems arising from RSFADEs, the Crank-Nicolson method is used to discretize the first order time partial derivative, while the finite volume method is adopted to approximate the first order space partial derivative of advection term and the Riesz space fractional partial derivative of the diffusion term. Furthermore, two main theoretical results about stability and convergence of the CN-FVM scheme are also discussed. It is proved that CN-FVM scheme is unconditionally stable and convergent with the accuracy of ${\mathcal O} (h^2+\tau^2)$ in the discrete $L_2$-norm, where $h$ and $\tau$ denote the spatial and temporal step sizes, respectively. Finally, some numerical experiments are presented to confirm the correctness of the theoretical analysis of the proposed scheme.

In this paper, a class of HIV time-delayed differential equation model with antiviral drug periodic treatment is established to study the threshold dynamics of~HIV~infection in the host. In order to investigate the influence of viral reverse transcription and budding processes on HIV infection process, we incorporate two time-dependent time delays into the model to characterize viral reverse transcription and budding periods respectively. Firstly, the well posedness of the model, including the global existence of periodic solutions and the dissipation of the system, is studied by using functional differential equation theory; Secondly, the functional expression of the basic reproduction number of the model $R_0$ is derived from the definition of the basic reproduction number for the periodic infectious disease compartment model; Finally, the global dynamic behavior of the system is discussed. More precisely, applying the uniform persistence theory, it is proved that HIV infection and replication in the host are persistent when $R_0>1 $, the infection free periodic solution of the system is globally attractive when $R_0<1$, that is, HIV will eventually be eliminated within the host.

Viral hepatitis C (simply referred to as hepatitis C) is a form of viral hepatitis caused by infection with the hepatitis C Virus (HCV). HCV will cause chronic inflammation, necrosis, and fibrosis of the liver, some patients may develop cirrhosis and hepatocellular carcinoma (HCC). In this paper we take advantage of the hepatitis data set to construct penalized trinomial logit models to diagnose the disease stages of patients. Firstly, we select 12 physiological indicators of patients as a predictor vector, and choose 3 disease stages of hepatitis C as the response variable. Secondly, we apply the 70% data as the training set to learn LASSO/Ridge/ENet penalized trinomial logit model, and take advantage of the coordinate descent algorithm to complete variable selection and obtain parameter estimations. Thirdly, we apply the remaining 30% data as the testing set, and combine three-class confusion matrix, the ROC (receiver operating characteristic) surface, HUM (hypervolume under the ROC manifold), PDI(polytomous discrimination index) to assess the prediction accuracy to disease stages. Finally, we introduce some machine learning methods such as artificial neural network (ANN), support vector machine (SVM) and random forest (RF)to compare with the proposed penalized trinomial logit models, and found that penalized trinomial logit models possess the best three-class prediction performance. They can not only improve the diagnostic accuracy to disease stages, but also reduce the cost of hepatitis C detection.

The research on customer strategy behavior in queueing system combined with game theory is a hot topic in the current queueing theory. This paper studies the strategic behavior of risk-sensitive customers in discrete-time queueing systems. Different from the classical economics of queues, the utility function in this paper is an expectation-variance quadratic utility function. Based on the Nash equilibrium and Markov process theory, we study the game behavior of Geo/Geo/1 queueing system with risk-sentitive customers under fully observable case and fully unobservable case, respectively. The individual optimal joining strategy, the joining strategy for the social net welfare and the server’s profit optimization are obtained. It is found that the smaller the risk sensitivity coefficient is, the more customers like to take risks and the stronger the willingness to join the system. Some numerical experiments are provided to illustrate the effect of the risk sensitivity coefficient on the customer strategic behavior.

The information collection capabilities in the era of big data have brought more complex data structures to time series analysis. Matrix-valued time series are common in the fields of macroeconomics, finance, and management, and are manifested as continuous observations of multiple indicators in multiple locations. The matrix autoregressive model is superior to the vector autoregressive model in terms of model expression and prediction due to its bilinear structure and fewer parameters. However, the matrix autoregressive model only contains the prediction structure of the time dimension, and does not have the prediction structure of the space dimension. For this reason, we added the spatial lag regression term containing the spatial weight matrix, and proposed the time-space lag regression model of matrix-valued time series. We assign scale parameters and adjustment parameters to each location and each variable to test whether the spatial prediction effect exists. Since the proposed model has no endogenous problem, this paper uses the partial iterative least squares method to obtain good parameter estimates. We also give the BIC criterion for model order selection and propose a model rank-reduced estimate. In addition, for the case of thick-tail distribution of the residual term, a robust estimation based on Huber loss function is proposed. With the increase of the sample size of the simulated data, the deviation and variance of the estimator tends to decrease gradually. The actual data shows that the proposed model has moderate model complexity and the smallest out-of-sample prediction error.

Preview control can improve control system performance by utilizing future desired ox external disturbance information, repetitive control introduces human learning mechanism into the control system. Therefore, more and more applications are seen in various real engineering fields, attracting a wide spread attention of researchers. For a class of uncertain system with time-varying delay, the problem of preview repetitive control (PRC) is proposed under the assumption that the reference signals are previewed and periodic. First, the repetitive controller is introduced and the error system in the preview control theory is adopted, and an two-dimensional (2D) augmented error system is constructed. This leads to the problem of preview repetitive controller’s design being transformed into a output feedback control problem of the augmented error system. Then, for the 2D augmented error system, the output equation is modified to fuse the previewed future information and the repetitive controller while considering output feedback. Based on Lyapunov method and LMI technique, the conditions of asymptotic stability of the closed-loop system and the design method of the preview repetitive controller are given. Finally, the effectiveness of the proposed method is illustrated by two simulation experiments.

There are almost no simple linear dynamic systems in nature, and most of them exist in the form of non-conservative nonlinear dynamic systems. Non-standard Lagrange functions can be used for dynamic modeling of non-conservative nonlinear problems. The fractional model is also a good choice for studying complex dynamics and physical behavior. Therefore, this paper studies the Noether symmetry and conserved quantity of non-standard Lagrange systems under generalized fractional operators. Firstly, the Lagrange equation of non-standard Lagrangian system under generalized operator is established. Then, based on the invariance of Hamilton action under infinitesimal transformation, the Noether theorem of non-standard Lagrangian system under generalized fractional operator is established, and the symmetry and corresponding conserved quantity of the system are given. Under certain conditions, the Noether conserved quantities of non-standard Lagrangian systems under generalized fractional operators can be reduced to the Noether conserved quantities of non-standard Lagrangian systems of integer order. Finally, examples are given to illustrate the specific application of the obtained results.

This paper focus on the multiplicity problem of positive solutions of integral boundary value problems for an impulsive fractional differential equation with positive homomorphism operator. By using the classical Guo-Krasnosel’skii fixed point theorem, some sufficient conditions for the existence of at least two positive solutions of impulsive fractional differential equation are derived. Finally, one example is shown to illustrate the theoretical results.

As an important part of the financial market system, it is very important for insurance companies to choose the optimal investment and reinsurance strategies. This paper considers an optimal reinsurance and investment problem for an insurance firm under the criterion of mean-variance. Assume that insurers diversify their risk by purchasing proportional reinsurance, which the surplus process is a diffusion process similar to the classical Cramér-Lundberg model. In addition, insurers increase their income by investing in risk-free assets and risky assets, which the prices of risky assets follow the Volterra Heston model. Due to the non-Markovian and non-semimingale properties of the Volterra Heston model, the classical stochastic optimal control framework is no longer applicable. By constructing an auxiliary stochastic process, we obtain the optimal investment and reinsurance strategies and the efficient frontier, which depend on the solution to a Riccati-Volterra equation. Finally, we numerically analysis the relationship among optimal strategies, effective frontier, volatility roughness and reinsurance factors. It is found that the rougher the volatility of stocks, the greater the demand of insurance companies for stocks and reinsurance.

The Riemann-Hilbert (RH) problem is performed to study the modified Korteweg-de Vries (mKdV) equation in this work, and we give an effective method to obtain the soliton solution with the rapidly decaying initial value. The important properties of Jost functions and scattering matrix are obtained by the direct scattering to construct a suitable RH problem, and then the relationship between the solution of the RH problem and the potential function is established. In the inverse problem, two cases of scattering data, including simple zeros and double zeros, are considered, and the corresponding RH problem is solved. Then, the general forms of solutions for mKdV equation in two cases are given successfully. Finally, in combination with specific parameters, the multi-soliton solutions image propagation in two cases are given in detail.

In this paper, we consider the method for optimization problem with $l_1-$norm. This kind of problem is widely used in compressed sensing and other fields. Based on the proposed smoothing function, a new conjugate gradient method to solve this optimization problem with $l_1-$norm is proposed in this paper. The global convergence of the proposed method is analyzed under general conditions, and the related numerical results also show the effectiveness of it.

Experiments are indispensable in many fields, for example, well-designed scientific experiments are needed in the research, development and testing of new products. When it is very difficult to change the level of factors in the experiment, how to arrange the run order reasonably is a very important problem. In this paper, some basic theories of run order with the minimum and maximum level changes are studied, and the optimal run order construction methods are discussed for full factorial designs, nonregular fractional factorial designs and uniform designs. As for some designs are widely used in practice, the run order with the minimum and maximum level changes and the corresponding of level changes are given by using the results of this paper.

The structural stability for the Brinkman-Forchheimer fluid interfacing with a Darcy fluid in a bounded region in $\mathbb{R}^3$was studied. We assumed that the velocity of fluid was slow and it was governed by the Brinkman-Forchheimer equations in $\Omega_1$, while in $\Omega_2$, we supposed that the saturated flow satisfies the Darcy equations. With the aid of the fourth norm estimates for the temperatures and the Sobolev inequality, we formulated an energy expression, and the expression satisfies a differential inequality. By integrating, we were able to demonstrate the continuous dependence result for the Brinkman coefficient.

The uniform designs are accepted widely because of its robust and easy to use, flexible characteristics. In order to distribute the points evenly in the experimental domain, many criteria ($L_2$-discrepancy) have been forwarded to measure the uniformity of the design array. At present, centered $L_2$-discrepancy, wrap-around $L_2$-discrepancy, mixed discrepancy and so on are widely used. Symmetric $L_2$-discrepancy has better geometric sense, but the poor performance at projection uniformity limits the use of SD. To refine the projection properties of SD, a projection weighted SD is proposed. The SD was exponentially weighted. The projection weighted SD can retain the excellent properties of the original discrepancy, and overcome the original defects effectively, and has better performance. The foldover is a useful technique in construction of factorial designs. In this paper, the projection weighted symmetric $L_2$-discrepancy is used as the optimality criterion to evaluate the quality of the foldover scheme. Lower bounds for projection weighted symmetric $L_2$-discrepancy on combined two-level U-type designs under a general foldover plan are obtained, which can be used as a benchmark for searching optimal foldover plans.

Considering the uncertainty of financial market data volatility, a new logmean reversion jump diffusion 4/2 random volatility (LMRJ-4/2-SV) model was proposed in this paper. Firstly, the LMRJ-4/2-SV model was constructed, and the European option pricing formula based on LMRJ-4/2-SV model was obtained by using FFT and other methods. Secondly, descriptive statistical analysis of the actual market data was carried out to discuss the price change characteristics of the underlying asset and the applicability of the LMRJ-4/2-SV model, and the model parameters were estimated by particle swarm optimization algorithm. Finally, European options were priced based on the option pricing formula and parameter estimates under the LMRJ-4/2-SV model, and the pricing results were compared with the 4/2, 3/2, Heston model estimates and market prices. The results show that the pricing error of European option based on LMRJ-4/2-SV model is minimal, and the pricing results have obvious advantages over other stochastic volatility models.

This study is concerned with numerical solutions of the two-dimensional Fisher-Kolmogorov-Petrovsky-Piscounov equation (Fisher-KPP) by a class of weighted structure-preserving finite difference methods (W-SP-FDMs) combined with Richardson extrapolation methods (REMs). By using the discrete energy analysis method, it is shown that as the parameters $\alpha$, $p$ and $\theta$, and the ratios of temporal meshsize to spatial meshsizes satisfy certain conditions, the current W-SP-FDMs possess many properties, such as, preserving positivity, preserving boundedness, preserving monotonicity, and has a convergence order of $O(\tau+h_x^2+h_y^2)$ in $L^{\infty}$- norm. Also, by using the discrete energy analysis method, it is shown that the REMs, which are developed by the asymptotic expansion formula of the numerical solutions, can make the final solutions convergent with an order of $O(\tau^{2}+h_{x}^{4}+h_{y}^{4})$ in $L^{\infty}$- norm, thus improving computational efficiency. Finally, numerical results confirm the correctness of theoretical findings and high performance of the current methods. It is worthwhile to mention that additional condition for the ratios of temporal meshsize to spatial meshsizes is not supplemented as REMs are used.

In this paper, we establish the optimality conditions for the nonsmooth multiobjective optimization problem by using Fréchet subdifferential and the extremal principle. First, the necessary conditions of semi-infinite multiobjective optimization problems are studied. And then, we establish the necessary condition of Henig proper efficient solution for the nonsmooth multiobjective optimization problem.

Due to the influence of observation noise, the identification test for volatility function of diffusion model will fail in high frequency environment. In this paper, we used local-averaging method to denoise the observed data. Based on the smoothing values, we combined the conditional moment and the nonparametric kernel estimation method to construct U statistics to identify the volatility function of the diffusion model. The test statistic converges to the standard normal distribution under the condition that the form of the volatility function is specified correctly. The Monte Carlo simulation results show that this statistic has more reasonable test size and stronger test power than the existing test methods. Using the constructed test statistic to identify the volatility function of the logarithmic price data of bank of China stock, more reasonable test results are obtained.

This paper studies the problem of testing the mean of high-frequency functional data. For functional data with infinite number of principal components and spiked eigenvalues of covariance operators, the classical Chi-square or mixed Chi-square test constructed based on the dimension reduction method using functional principal components will become invalid due to insufficient sample size and strong conditions of covariance operators. Therefore, this paper proposes a randomized test to solve this problem, and proves the large sample properties. Further, the numerical simulation of limited samples is used to verify the effectiveness of the proposed test. Finally, this method is applied to the phoneme data.

Blow up phenomena of solutions to a dissipative generalized Tricomi equation with variable coefficients and nonlinearity of derivative type in the subcritical case are considered. By constructing some time-dependent functionals associated with test function methods and Bessel equations, an iterative frame and the first lower bound of the time-dependent functional are obtained. Then, blow-up of solutions and upper bound estimate for the lifespan to the Cauchy problem are proved via iteration arguments.

In this paper, the general nonlinear moisture migration equations are studied with a mixed finite element method. The superconvergence of the equations is proved by use of bilinear element $Q_{11}$ and zero order Raviart-Thomas element ($Q_{10}\times Q_{01}$). With the help of the interpolation operators of the above two elements and mean-value technique, the superconvergence results of order $O(h^2)$ are obtained for the semi-discrete scheme of the equations. For the linearized Crank-Nicolson (C-N) fully-discrete scheme, the superconvergence results of order $O(h^2+\tau^2)$ are also derived, here $h$ is the subdivision parameter, $\tau$ is the time step. This method shows that if the linearization problem has superconvergence, the corresponding nonlinear problem has the same superconvergence. Finally, a numerical example is provided to illustrate the correctness of the theoretical analysis and the feasibility of the proposed method.

A 3-rainbow dominating function of a graph $G$ is a mapping $f$ from the vertex set $V$ of $G$ to the power set of the set $\{1,2,3\}$ such that any vertex $v$ with $f(v)=\varnothing$ satisfies that $\bigcup\limits_{u\in N(v)}f(u)=\{1,2,3\}$, where $N(v)$ is the neighborhood of $v$. The weight of a 3-rainbow dominating function $f$ of $G$ is $\sum\limits_{v\in V}|f(v)|.$ If $f$ is a 3-rainbow dominating function of a graph $G$ and its complement, then $f$ is called a global 3-rainbow dominating function of $G$. The global 3-rainbow domination number of $G$ is the minimum weight of a global 3-rainbow dominating function of $G$. By analyzing the structure of graphs, using the method of categorical discussion, we completely characterize the graphs with global 3-rainbow domination number equal to the number of vertices.

An adjacent vertex distinguishing edge coloring of graph $ G $ is a proper edge coloring of $ G $ such that any pair of adjacent vertices have distinct sets of colors. The mininum number of colors required for an adjacent vertex distinguishing edge coloring of $ G $ is denoted by $ \chi_{a}{'}(G) $. This paper proves that $ G $ is a normal planar graph with girth at least $ 6 $, $ \chi_{a}{'}(G) \leq \max\{6, \Delta(G)+1\} $.

Under investigation in this work is the integrable coupled Sasa-Satsuma equation, which can be used to describe the propagation dynamics of two ultrashort pulses in the birefringent or two-mode fiber. Through the Darboux-dressing transformation, we obtain a family of semirational solutions. This family of solutions exhibits various scenes of superimposition between rogue waves and breathers. These results may contribute to enriching and explaining some related nonlinear phenomena in optical fibers and dispersive media.

In this paper, we mainly introduce a topological definition of quasi-shadowing property with respect to a leaf-set (or a foliation) on a dynamical system, which is equivalent to the general definition of quasi-shadowing property with respect to a foliation on Riemannian manifolds. At the same time, we discuss the relationship between the quasi-shadowing property with respect to a leaf-set (or a foliation) and an inverse limit of a sequence of shifts of finite type on the compact Hausdorff totally disconnected spaces and Riemannian manifolds respectively. In addition, we give a sufficient condition for the factor maps to preserve quasi-shadowing property.

In this paper, we study the 1-dimensional linear Navier-Stokes-Fourier equations and obtain the pointwise estimates of the decay properties of the solution under the appropriate initial value conditions, and describe the decay direction of the solution, and verify that the generalized Huygens’ principle holds. To this end, we divide the Fourier transform of the Green function of the equations into low-frequency, mid-frequency and high-frequency parts by means of Fourier transform, and prove the decay properties of the Green function in the corresponding frequency parts, and then obtain the decay estimates of the solutions by means of the Fourier inverse transform and the properties of the fundamental solutions.

Our primary objective of this article is to study a class of nonlinear telegraph systems with a parameter. Using index theory of fixed points for compact operators, some new criteria for the existence, multiplicity and nonexistence of positive doubly periodic solutions are established in terms of different values of parameter.

In this paper, it is proved that if a dynamical system has the periodic $\mathscr{M}_{\alpha}$-shadowing property or periodic $\mathscr{M}^{\alpha}$-shadowing property, then the dynamical system restricted on its measure center has the same shadowing property. Conversely, if a dynamical system restricted on its measure center has the periodic $\mathscr{M}_{\alpha}$-shadowing property (resp., periodic $\mathscr{M}^{\alpha}$-shadowing property), then the dynamical system has the periodic $\mathscr{M}_{\beta}$-shadowing property (resp., periodic $\mathscr{M}^{\beta}$-shadowing property) for any $\beta\in [0, \alpha)$. Moreover, it is obtained that for an equicontinuous system, many shadowing properties are equivalent to the condition that it has a trivial measure center.

The Kelvin-Voigt fluid passing through a semi infinite cylinder is considered, in which the Kelvin-Voigt fluid satisfies the homogeneous boundary condition on the side of the cylinder. Using the methods of energy estimation and a priori estimations, the exponential decay property of strain energy with distance from the finite end of the cylinder is proved. This type of result can be regarded as the "distance effect" of Saint-Venant principle.

Under widely orthant dependent samples, two kinds of risk measure are considered. The Bahadur representation and strong consistency of the quantile estimator for VaR are discussed. And the strong consistency and its rate of the CVaR estimator are established, by the suitable choice of some constants, their rates are near $O(n^{-\frac{1}{2}})$. In order to better illustrate performances of the VaR and CVaR estimators, we conduct numerical simulations under some WOD sequences by ARMA(1,1) and MA(1) models, and discover that the estimators are high performance by tables of the exact and estimated values of VaR and CVaR, and their curve figures.

In this paper we consider an $M/G/1$ repairable queueing system with Bernoulli maintenance and admission control policy, in which whenever the system becomes empty, the manager shuts down the system immediately with probability $p(0\leq p\leq1)$ and arranges the repairman to maintenance it with a random time length, and at most $M(\geq1)$ customers are allowed to enter into the system during its maintenance period. Otherwise, the manager does not shut down the system with probability $(1-p)$ and the customers who arrive at the system will be served at once until the system becomes empty again. Applying the total probability decomposition, renewal process theory and Laplace transform tool, we study the transient distribution of queue size at any time $t$ in detail,and obtain the expressions of the Laplace transform of the transient queue-length distribution. Furthermore, the explicit recursive formulas of the steady-state distribution of the queue size at any time $t$ are derived by using L'Hospital's rule. Meanwhile, the expressions for the probability generating function of the steady-state queue-length distribution and the expected queue size are presented. Finally, we employ the renewal reward theory to derive the expression of the long-run expected cost per unit time of the system. Moreover, numerical examples are presented to discuss the one-dimensional optimal control strategy for minimizing the expected cost of the system as well as the two-dimensional optimal control strategy.

In this paper, the Lévy process is used to model the surplus level of an insurance company. It is assumed that the insurance company observes the surplus level at a fixed time and makes a decision after each observation. If the observed surplus level is less than a given critical level and is non-negative, the shortfall is injected once to bring the surplus level back to the critical level. If the observed surplus level is negative, ruin is declared immediately. Using the Fourier cosine series expansion method, we propose some numerical methods for computing the finite-time expected total discounted cost of capital injections before ruin and the finite-time expected discounted penalty function. The accuracy and efficiency of the method are demonstrated through error analysis and numerical examples, and the effect of each parameter on the results is studied.

We consider the existence of weak solutions to a class of elliptic equations with nonstandard growth condition and zero order term. The main tools are the weak convergence method for PDEs and Young measure method. The perturbed problem is treated as a starting point. In view of the assumptions on the zero order term and the integrability of the source term, we pick up some appropriate test functions, which are vital to some necessary a priori estimates and the limit process. Thanks to the Young measure method, the weak limit of the nonlinear term is identified. The existence of a weak solution is obtained by passage of the limit.

In this work, aiming at the problem of lacking analytical methods and it is typically difficult to find explicit exact solutions to the population balance equations with fragmentation processes. We consider the admitted Lie groups, group invariant solutions, reduced integro-ordinary differential equations and explicit exact solutions of two classes of integro-partial differential equations (population balance equations with fragmentation processes) by use of the method of scaling transformation group analysis, the method of classical Lie group analysis and the method of developed Lie group analysis. Firstly, the admitted scaling transformation Lie groups of the integro-partial differential equations are explored by using the method of scaling transformation group analysis. Secondly, the integro-partial differential equations are transformed into pure partial differential equations, the admitted Lie groups of the pure partial differential equations are calculated by use of the methods of classical Lie group analysis. Thirdly, the admitted Lie groups of the original integro-partial differential equations are determined by use of the methods of developed Lie group analysis combining with the related results obtained by method of scaling transformation group and the method of classical Lie group analysis. Finally, the admitted Lie groups of the original integro-partial differential equations are successfully found. All group invariant solutions, reduced integro-ordinary differential equations and explicit exact solutions are given. The related analysis for dynamic behavior characteristics of a solution with evolution of the size distribution are also presented.