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This paper aims to introduce a new numerical approach based on the local weak form and the Petrov–Galerkin idea to numerically simulation of a predator–prey system with two-species, two chemicals and an additional chemotactic influence.

In the first proceeding, the space derivatives are discretized by using the direct meshless local Petrov–Galerkin method. This generates a nonlinear algebraic system of equations. The mentioned system is solved by using the Broyden’s method which this technique is not related to compute the Jacobian matrix.

This current work tries to bring forward a trustworthy and flexible numerical algorithm to simulate the system of predator–prey on the nonrectangular geometries.

The proposed numerical results confirm that the numerical procedure has acceptable results for the system of partial differential equations.

The purpose of this study is to examine the effects of using discrete and continuous porous layers on the convective heat transfer improvement for multiple slot jet impingement onto a flat surface under magnetic field.

In the domains which are separated by the porous layers, uniform magnetic field with different strengths is used and as the solution technique finite element method is used. The numerical study is conducted considering different values of parameters: Reynolds number (250–1000), strength of magnetic field in different domains (Hartmann number between 0 and 20), permeability of discrete or continuous layers (Darcy number between 105 and 102) and number of layers in discrete case (2–10). Artificial neural network is used for performance estimation of systems equipped with different types of porous layers.

It is observed that significant differences occur in the local Nu between the discrete and continuous layer case, especially at lower Re, while peak Nu value is 77% higher in discrete layer configurations as compared to continuous one at Re = 250. Upper domain magnetic field results in average Nu enhancement, while the trend is opposite for the lower domain magnetic field strength. The increment amount becomes 10%, while the reduction amount is obtained as 38% at the highest magnetic field strengths. The permeability of layers in both cases and number of layers in discrete porous layer case provide effective solution for the cooling performance control. A modeling approach based on artificial neural networks provides fast thermal performance estimations of multiple impinging jets equipped with discrete and continuous porous layers.

Outcomes of the study are useful in development and optimization of new cooling systems in many thermal engineering systems encountered in photovoltaic panels, micro-electro-mechanical systems, metal processing and many others.

The space fractional PDEs (SFPDEs) play an important role in the fractional calculus field. Proposing a high-order, stable and flexible numerical procedure for solving SFPDEs is the main aim of most researchers. This paper devotes to developing a novel spectral algorithm to solve the FitzHugh–Nagumo models with space fractional derivatives.

The fractional derivative is defined based upon the Riesz derivative. First, a second-order finite difference formulation is used to approximate the time derivative. Then, the Jacobi spectral collocation method is employed to discrete the spatial variables. On the other hand, authors assume that the approximate solution is a linear combination of special polynomials which are obtained from the Jacobi polynomials, and also there exists Riesz fractional derivative based on the Jacobi polynomials. Also, a reduced order plan, such as proper orthogonal decomposition (POD) method, has been utilized.

A fast high-order numerical method to decrease the elapsed CPU time has been constructed for solving systems of space fractional PDEs.

The spectral collocation method is combined with the POD idea to solve the system of space-fractional PDEs. The numerical results are acceptable and efficient for the main mathematical model.