Iranian Journal of Numerical Analysis and Optimization
Volume:13 Issue: 1, Winter 2023

The meshless Fragile Points method (FPM) is applied to find the numerical solutions of the Schrödinger equation on arbitrary domains. This method is based on Galerkin’s weak-form formulation, and the generalized finite difference method has been used to obtain the test and trial functions. For partitioning the problem domain into subdomains, Voronoi diagram has been applied. These functions are simple, local, and discontinuous poly-nomials. Because of the discontinuity of test and trial functions, FPM may be inconsistent. To deal with these inconsistencies, we use numerical flux corrections. Finally, numerical results are presented for some exam-ples of domains with different geometric shapes to demonstrate accuracy, reliability, and efficiency.

To solve challenges occurred in the existence of large sets of data, recent improvements of machine learning furnish promising results. Here to pro-pose a tool for predicting lesser liquid credit default swap (CDS) rates in the presence of CDS spreads over a large period of time, we investigate different machine learning techniques and employ several measures such as the root mean square relative error to derive the best technique, which is useful for this type of prediction in finance. It is shown that the nearest neighbor is not only efficient in terms of accuracy but also desirable with respect to the elapsed time for running and deploying on unseen data.

Like the Polak-Ribi`ere-Polyak (PRP) and Hestenes-Stiefel (HS) meth-ods, the classical Liu-Storey (LS) conjugate gradient scheme is widely be-lieved to perform well numerically. This is attributed to the in-built capa-bility of the method to conduct a restart when a bad direction is encoun-tered. However, the scheme’s inability to generate descent search direc-tions, which is vital for global convergence, represents its major shortfall. In this article, we present an LS-type scheme for solving system of mono-tone nonlinear equations with convex constraints. The scheme is based on the approach by Wang et al. (2020) and the projection scheme by Solodov and Svaiter (1998). The new scheme satisfies the important condition for global convergence and is suitable for non-smooth nonlinear problems. Fur-thermore, we demonstrate the method’s application in restoring blurry im-ages in compressed sensing. The scheme’s global convergence is established under mild assumptions and preliminary numerical results show that the proposed method is promising and performs better than two recent meth-ods in the literature.

A mathematical collocation solution for generalized Burgers–Huxley and generalized Burgers–Fisher equations has been monitored using the weighted residual method with Hermite splines. In the space direction, quintic Hermite splines are introduced, while the time direction is dis-cretized using a finite difference approach. The technique is determined to be unconditionally stable, with order (h4 + △t) convergence. The tech-nique’s efficacy is tested using nonlinear partial differential equations. Two problems of the generalized Burgers–Huxley and Burgers–Fisher equations have been solved using a finite difference scheme as well as the quin-tic Hermite collocation method (FDQHCM) with varying impacts. The FDQHCM computer codes are written in MATLAB without transforming the nonlinear term to a linear term. The numerical findings are reported in weighted norms and in discrete form. To assess the technique’s appli-cability, numerical and exact values are compared, and a reasonably good agreement is recognized between the two.

We suggest an a priori method by introducing the concept of AP - equitable efficiency. The preferences matrix AP , which is based on the partition P of the index set of the objective functions, is given by the decision-maker. We state the certain conditions on the matrix AP that guarantee the preference relation ⪯eAP to satisfy the strict monotonicity and strict P -transfer principle axioms. A problem most frequently encountered in multiobjective optimization is that the set of Pareto optimal solutions provided by the optimization pro-cess is a large set. Hence, the decision-making based on selecting a unique preferred solution becomes difficult. Considering models with Ar P -equitable efficiency and A∞ P -equitable efficiency can help the decision-maker for over-coming this difficulty, by shrinking the solution set.

One of the major problems in applied mathematics and engineering sciences is solving nonlinear equations. In this paper, a family of eight-order interval methods for computing rigorous bounds on the simple zeros of nonlinear equations is presented. We present the convergence and er-ror analysis of the introduced methods. Also, the introduced methods are compared with the well-known interval Newton method and interval Ostrowski-type methods. Finally, we propose a technique based on the combination of the newly introduced approach with the extended interval arithmetic to find all of the roots of a nonlinear equation that are located in an initial interval.

A new numerical scheme based on Genocchi polynomials is constructed to solve fractional Sturm–Liouville problems of order two in which the fractional derivative is considered in the Caputo sense. First, the differen-tial equation with boundary conditions is converted into the corresponding integral equation form. Next, the fractional integration and derivation op-erational matrices for Genocchi polynomials, are introduced and applied for approximating the eigenvalues of the problem. Then, the proposed polynomials are applied to approximate the corresponding eigenfunctions. Finally, some examples are presented to illustrate the efficiency and accu-racy of the numerical method. The results show that the proposed method is better than some other approximations involving orthogonal bases.

This paper focuses on the result of inclined angle on bioconvection of porous media bounded by cavity wall square enclosure filled with both nanofluid and gyrotactic microorganisms passing through the media with pores. The dimensionless velocity, temperature, concentration, and mass transformation equations are solved by using the weighted residual Galerkin’s finite element method. The result of the inclination angle from δ = 0◦ to δ = 180◦ in a square cavity is interpreted. The outcomes of incli-nation on various key parameters, such as Rayleigh number, bioconvective Rayleigh number, Peclet number, Brownian motion, and the ratio of buoy-ancy, are discussed. Furthermore, the mean Nusselt number, Sherwood number, and density number are discussed at vertical walls.