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The key idea and contribution of this study are to present the innovative functional matrix approach for solving the two chemical, mathematical problems such as the absorption of $$C{O}_{2}$$ C O 2 into phenyl glycidyl ether (PGE) and the chemical kinetic problem. These chemical problems are characterized as a system of nonlinear ODEs with initial and boundary conditions. Numerical outcomes are obtained to establish the simplicity and efficacy of the developed scheme. Graphs and tables show how consistently and effectively the developed strategy works. Results obtained show that the newly adopted technique is more precise and effective than other methods such as Adam–Bashforth–Moultan method (ABM), Runge–Kutta method (RK4), Adomian Decomposition method (ADM), and residual method (RM). Mathematical software called Mathematica 11.3 has been used to perform all calculations. Theorems explain the convergence of this approach.

This study proposed an efficient numerical technique for coupled differential equations (CDEs) using the clique polynomials of the Complete graph. Recently, Graph theory has dragged the attention of many mathematicians due to its wide applications. Here, some problems have been considered to examine the proposed scheme proficiency. Some theorems on convergence are discussed. Here, the CDEs are rehabilitated into an algebraic equation system using the clique polynomials and collocation technique. The proposed scheme results are compared with the undetermined coefficients (UDCM) and variation of parameters method (VPM) solutions through tables and graphs. Obtained results reveal that the current approach is more accurate.

We report a case of surgically treated splenic abscess in a HIV patient. The patient presented with fever and splenomegaly. The diagnosis was based on ultrasonography and computed tomography. Surgical experience with clinically overt splenic abscess remains limited more so in a HIV patient.