In this paper, our purpose is to study the classical continuous optimal control (CCOC)  for quaternary nonlinear parabolic boundary value problems (QNLPBVPs). The existence and uniqueness theorem (EUTh) for the quaternary state vector solution (QSVS) of the weak form (WF) for the QNLPBVPs with a given quaternary classical continuous control vector (QCCCV) is stated and proved via the Galerkin Method (GM) and the first compactness theorem under suitable assumptions(ASSUMS). Furthermore, the continuity operator for the existence theorem of a QCCCV dominated by the QNLPBVPs is stated and proved under suitable conditions.

In this paper, three approximate methods namely the Bernoulli, the Bernstein, and the shifted Legendre polynomials operational matrices are presented to solve two important nonlinear ordinary differential equations that appeared in engineering and applied science. The Riccati and the Darcy-Brinkman-Forchheimer moment equations are solved and the approximate solutions are obtained. The methods are summarized by converting the nonlinear differential equations into a nonlinear system of algebraic equations that is solved using Mathematica®12. The efficiency of these methods was investigated by calculating the root mean square error (RMS) and the maximum error remainder (𝑀𝐸𝑅n) and it was found that the accuracy increases with increasi

This paper is attempt to study the nonlinear second order delay multi-value problems. We want to say that the properties of such kind of problems are the same as the properties of those with out delay just more technically involved. Our results discuss several known properties, introduce some notations and definitions. We also give an approximate solution to the coined problems using the Galerkin's method.

Solutions of dyes Rhodamine 6G (Rh6G) and Coumarin480(C480) were prepared at five concentrations (1x10-3, 5x10-4, 1x10-4, 5x10-5 and1x10-5) mol/l, the mixing was stirred to obtain on a homogenous solution, the(poly methyl-methacrylate) (PMMA) was solved by chloroform solvent with certain ratio, afterward (PMMA+Rh6G) and (PMMA+C480) thin films were prepared by casting method on glass block which has substrate with dimensions (7.5 x2.5)cm2, the prepared samples were left in dark place at room temperature for 24 hours to obtain uniform and homogenous thin films. UV-VIS absorption spectra, transmission spectra and fluorescence spectra were done to measure linear refractive index and linear absorption coefficient. The nonlinear optical proper

Background: Ulcerative colitis (UC) is an inflammatory bowel disease restricted to the large intestine, characterized by superficial ulceration. It is a progressive and chronic disease requiring long-term treatment. Although its etiology remains unknown, it is suggested that environmental factors influence genetically susceptible individuals, leading to the onset of the disease. (C-X-C) ligand 9 is a chemokine that belongs to the CXC chemokine family, it plays a role in the differentiation of immune cells such as cytotoxic lymphocytes, natural killer T cells, and macrophages. Its interaction with its corresponding receptor CXCR3 which is expressed by a variety of cells such as effector T cells, CD8+ cytotoxic T cells, and macrophage