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In this paper, we present an improved imperialist competitive algorithm for solving an inverse form of the Huxley equation, which is a nonlinear partial differential equation. To show the effectiveness of our proposed algorithm, we conduct a comparative analysis with the original imperialist competitive algorithm and a genetic algorithm. The improvement suggested in this study makes the original imperialist competitive algorithm a more powerful method for function approximation. The numerical results show that the improved imperialist competitive algorithm is an efficient algorithm for determining the unknown boundary conditions of the Huxley equation and solving the inverse form of nonlinear partial differential equations.

In this paper, we develop a least-squares support vector machine (LS-SVM) for solving a nonlinear fractional-order Volterra’s population model in a closed system. The fractional rational Legendre functions with an orthogonal property on a semi-infinite domain have been used as the kernel of LS-SVM. Learning the solution is done by solving a non-linear constrained optimization problem. To accelerate the learning process, we propose two different approaches based on the orthogonality of kernels and a shared-memory task parallelization scheme for multi-core systems. By carrying out several experiments, it is seen that the proposed approaches provide accurate solutions for fractional-order Volterra’s population model.

The Lane-Emden type equations are employed in the modeling of several phenomena in the areas of mathematical physics and astrophysics. In this paper, a new numerical method is applied to investigate some well-known classes of Lane-Emden type equations which are non-linear ordinary differential equations on the semi-infinite domain [0,∞). We will apply a meshless method based on radial basis function differential quadrature (RBF-DQ) method. In RBFs-DQ, the derivative value of function with respect to a point is directly approximated by a linear combination of all functional values in the global domain. The main aim of this method is the determination of weight coefficients. Here, we concentrate on Gaussian (GS) as a radial function for approximating the solution of the mentioned equation. The comparison of the results with the other numerical methods shows the efficiency and accuracy of this method. Keywords: Radial basis functions, RBFs-DQ, Lane-Emden type equations, Gaussian, Semi-infinite

In this paper, we introduce an improved water strider algorithm designed to solve the inverse form of the Burgers-Huxley equation, a nonlinear partial differential equation. Additionally, we propose a physics-informed neural network to address the same inverse problem. To demonstrate the effectiveness of the new algorithm and conduct a comparative analysis, we compare the results obtained using the improved water strider algorithm against those derived from the original water strider algorithm, a genetic algorithm, and a physics-informed neural network with three hidden layers. Solving the inverse form of nonlinear partial differential equations is crucial in many scientific and engineering applications, as it allows us to infer unknown parameters or initial conditions from observed data. This process is often challenging due to the complexity and nonlinearity of the equations involved. Meta-heuristic algorithms and neural networks have proven to be effective tools in addressing these challenges. The numerical results affirm the efficiency of our proposed method in solving the inverse form of the Burgers-Huxley equation. The best results were obtained using the improved water strider algorithm and the physics-informed neural network with 10,000 iterations. With this iteration count, the mean absolute error of these algorithms is O ( 10 - 4 ) . Additionally, the improved water strider algorithm is nearly four times faster than the physics-informed neural network. All algorithms were executed on a computing system equipped with an Intel(R) Core(TM) i7-7500U processor and 12.00 GB of RAM, and were implemented in MATLAB., Competing Interests: Declarations. Competing interests: The authors declare no competing interests., (© 2024. The Author(s).)