Abstract: To retain intact state information of the district heating network (DHN) as possible in the optimization of heat and electricity integrated energy system (HE-IES), this article combines the transient heat flow and steady-state electric power flow to formulate the dynamic optimal energy flow (OEF) model of HE-IES. For efficient and standardized solution, the finite difference method is applied to convert the partial differential equation constraint (introduced by the temperature dynamics in DHN) into a set of linear equality constraints. The structure of applicable difference schemes for system optimization is analyzed, based on which, a scheme with balanced performances in stability, convergence, simulation accuracy, and computation burden is developed. Moreover, with the proposed multi-objective optimization based method to select proper spatial and temporal calculation step sizes, a compromise between model precision and solution complexity can be reached. Case studies validate the feasibility and benefits of our proposed dynamic OEF computing method, which can further provide support for making optimal planning and operating strategies.

Abstract: The nonlinear Sine-Gordon equation is one of the widely used partial differential equations that appears in various sciences and engineering. The main purpose of writing this article is providing an efficient numerical method for solving two-dimensional (2D) time-fractional stochastic Sine–Gordon equation on non-rectangular domains. In this method, radial basis functions (RBFs) and finite difference scheme are used to calculate the approximate solution of the mentioned problem. The complexity of solving this problem arises from its high dimension, irregular area, stochastic and fractional terms. Finite difference technique is applied to overcome on the problem dimension, whereas interpolation method based on RBFs is the best idea for solving problems defined in irregular domains. The stochastic Sine–Gordon equation is transformed into elliptic stochastic differential equations using the finite difference method and meshfree method based on RBFs are used to approximate the obtained stochastic differential equation. Some numerical examples are included to investigate the efficiency and accuracy of the presented method.

Abstract: The ongoing study deals with various forms of solutions for the biological population model with a novel beta-time derivative operators. This model is very conducive to explain the enlargement of viruses, parasites and diseases. This configuration of the aforesaid classical scheme is scouted for its new solutions especially in soliton shape via two of the well known analytical strategies, namely: the extended Sinh-Gordon equation expansion method (EShGEEM) and the Expa function method. These soliton solutions suggest that these methods have widened the scope for generating solitary waves and other solutions of fractional differential equations. Different types of soliton solutions will be gained such as dark, bright and singular solitons solutions with certain conditions. Furthermore, the obtained results can also be used in describing the biological population model in some better way. The numerical solution for the model is obtained using the finite difference method. The numerical simulations of some selected results are also given through their physical explanations. To the best of our knowledge, No previous literature discussed this model through the application of the EShGEEM and the Expa function method and supported their new obtained results by numerical analysis.

Abstract: This paper is concerned with numerical solution of time fractional stochastic advection-diffusion type equation where the first order derivative is substituted by a Caputo fractional derivative of order $$\alpha $$ ( $$0 <\alpha \le 1$$ ). This type of equations due to randomness can rarely be solved, exactly. In this paper, a new approach based on finite difference method and spline approximation is employed to solve time fractional stochastic advection-diffusion type equation, numerically. After implementation of proposed method, the under consideration equation is transformed to a system of second order differential equations with appropriate boundary conditions. Then, using a suitable numerical method such as the backward differentiation formula, the resulting system can be solved. In addition, the error analysis is shown in some mild conditions by ignoring the error terms $$O(\Delta t^2)$$ in the system. In order to show the pertinent features of the suggested algorithm such as accuracy, efficiency and reliability, some test problems are included. Comparison achieved results via proposed scheme in the case of classical stochastic advection-diffusion equation ( $$\alpha =1$$ ) with obtained results via wavelets Galerkin method and obtained results for other values of $$\alpha $$ with the values of exact solution confirm the validity, efficiency and applicability of the proposed method.

Abstract: In this work, two approaches have been introduced for solving a linear damped nonlinear Schrodinger equation (NLSE) for modelling the dissipative rogue waves (DRWs) and dissipative breathers (DBs). It is known that the linear damped NLSE is considered a non-integrable differential equation. Thus, it does not support an explicit analytic solution till now due to the presence of the linear damping term. Consequently, two accurate solutions will be derived and obtained in details. The first solution is called a semi-analytical solution while the second one is approximate numerical solution. In the two solutions, the analytical solution of the standard NLSE (i.e., in the absence of the damping term) will be used as initial solution for solving the linear damped NLSE. With respect to the approximate numerical solution, the moving boundary method (MBM) with the help of finite differences method (FDM) will be devoted for this purpose. The maximum residual (local and global) errors formula for the semi-analytical solution will be derived and obtained. Also, the numerical values of both the maximum residual local and global errors of the semi-analytical solution will be estimated using some physical data. Moreover, the error functions related to the local and global errors of the semi-analytical solution will be evaluated via using the nonlinear polynomial based on Chebyshev approximation technique. Furthermore, a comparison between the approximate analytical and numerical solutions will be carried out to check the accuracy of the two solutions. As a realistic application to some physical results; the obtained solutions will be used to investigate the characteristics of the dissipative rogue waves (DRWs) and dissipative breathers (DBs) in a collisional unmagnetized pair-ion plasma. Finally, this study help us to interpret and understand the dynamic behavior of modulated structures in various plasma models, fluid mechanics, optical fiber, Bose-Einstein condensate, etc.

Abstract: The thermodynamic features of the Reiner-Rivlin nanoliquid flow induced by a spinning disk are analyzed numerically. The non-homogeneous two-phase nanofluid model is considered to analyze the effect of nanoparticles on the thermodynamics of the Reiner-Rivlin nanomaterial, which also includes a temperature-dependent heat source (THS) and an exponential space-dependent heat source (ESHS). Further, the transfer of heat and mass is analyzed with velocity slip, volume fraction jump, and temperature jump boundary conditions. The finite difference method-based routine is used to solve the complicated differential equations formed after using the von-Karman similarity technique. Limiting cases of the present problem are found to be in good agreement with benchmarking studies. The relationship of the pertinent parameters with the heat and mass transport is scrutinized using correlation, which is further evaluated based on the probable error estimates. Multivariable models are fitted for the friction factor at the disk and heat transport, which accurately predict the dependent variables. The Reiner-Rivlin nanoliquid temperature is influenced comparatively more by the ESHS than by the THS. The Nusselt number is decreased by the ESHS and THS, whereas the friction factor at the disk is predominantly decremented by the wall roughness aspect. The increment in the non-Newtonian characteristic of the liquid leads more fluid to drain away in the radial direction far from the disk compared with the fluid nearby the disk in the presence of the centrifugal force during rotation. The increased thermal and volume fraction slip lowers the nanoliquid temperature and nanoparticle volume fraction profiles.

Abstract: In this study, a novel meshless stable numerical solver is proposed to solve the non-conservative form of shallow water equations. Since they form a hyperbolic system of equations, discontinuous solutions are allowed to transmit during the simulation. The generalized finite difference-split coefficient matrix method, recently proposed, is applied and improved using the flux limiter to eliminate the possible-appearing numerical oscillations. In the proposed scheme, the split-coefficient matrix method is adopted to convert the shallow water equations to the characteristic form. Then, the generalized finite difference method and the second-order Runge-Kutta method are employed for spatial and temporal discretization, respectively. The upwinding spatial derivatives can be approximated at every node using the half-disk shape of the star and generalized finite difference method. Applying the flux limiter technique, the expressions can automatically switch the proper discrete order when facing discontinuous solutions. Although the limiter function required the derivatives of different orders, the generalized finite difference method can solve these necessary expressions using the first- and second-order Tayler series. Several numerical examples are provided to demonstrate the capability of the proposed scheme, and the results are compared with other numerical schemes to show the effectiveness of the proposed generalized finite difference-flux limiter method.

Abstract: This paper deals with numerical treatment of singularly perturbed parabolic differential equations having large time delay. The highest order derivative term in the equation is multiplied by a perturbation parameter , taking arbitrary value in the interval . For small values of , solution of the problem exhibits an exponential boundary layer on the right side of the spatial domain. The properties and bounds of the solution and its derivatives are discussed. The considered singularly perturbed time delay problem is solved using the Crank-Nicolson method in temporal discretization and exponentially fitted operator finite difference method in spatial discretization. The stability of the scheme is investigated and analysed using comparison principle and solution bound. The uniform convergence of the scheme is discussed and proven. The formulated scheme converges uniformly with linear order of convergence. The theoretical analysis of the scheme is validated by considering numerical test examples for different values of .