The present research deals with the extremely sensitive temperature-sensing capabilities of defective one-dimensional photonic crystal structures (Si/PS/SiO₂). The proposed structure is realized by putting a defective layer of material silicon Dioxide (SiO₂) in the middle of a structure consisting of alternating layers of silicon (Si) and porous silica (PS). The transfer matrix method has been employed to examine the transmission characteristics of the proposed defective one-dimensional photonic crystal in addition to MATLAB software. The transmission spectra of the proposed structure in the visible light domain are computed throughout a temperature range of 25–900 °C, and we study the thermal properties related to the defective mode. Additionally, the impacts of changing the defect layer's thickness are examined. Due to the effects of thermal expansion and the thermo-optical coefficient, the defect mode varies significantly as the temperature increases. Our investigation shows that the proposed structure considerably impacts the transmission intensity of the defective mode. The theoretically obtained numeric values of the quality factor and sensitivity are 2216.6 and 0.085 nm/°C, respectively. The challenges presented by conventional temperature sensors could be overcome by the suggested defective photonic crystal sensor. These results are enough to support our claim that the present design can be used as an ultra-sensitive temperature sensor.

We investigate the numerical solution of linear fractional integro‐differential equations by least squares method with aid of shifted Chebyshev polynomial. Some numerical examples are presented to illustrate the theoretical results.

In this article, a novel and efficient approach based on Lucas polynomials is introduced for solving three-dimensional mixed Volterra–Fredholm integral equations for the two types (3D-MVFIEK2). This method transforms the 3D-MVFIEK2 into a system of linear algebraic equations. The error evaluation for the suggested scheme is discussed. This technique is implemented in four examples to illustrate the efficiency and fulfillment of the approach. Examples of numerical solutions to both linear and nonlinear integral equations were used. The Lucas polynomial method and other approaches were contrasted. A collection of tables and figures is used to present the numerical results. We observe that the exact solution differs from the numerical solution if the exact solution is an exponential or trigonometric function, while the numerical solution is the same when the exact solution is a polynomial. The Maple 18 program produced all of the results.

The main purpose of this article is to present a new technique for solving (1+1) mixeddimensional difference integro-differential Equations (2D-MDeIDEs) in position and time with coefficients of variables under mixed conditions. The equations proposed for the solution represent a link between time and delay in position that has not been previously studied. Therefore, the authors used the technique of separation of variables to transform the 2D-MDeIDE into one-dimensional Fredholm difference integro-differential Equations (FDeIDEs), and then using the Bernoulli polynomial method (BPM), we obtained a system of linear algebraic equations (SLAE). The other aspect of the technique of separation of variables is explicitly obtaining the necessary and appropriate time function to obtain the best numerical results. Some numerical experiments are performed to show the simplicity and efficiency of the presented method, and all results are performed by Maple 18.