Nonlinear behaviour of various problems is described by the Duffing model interpreted as a forced oscillator with a spring which has restoring force. In this paper, a new numerical approximation technique based on differential transform method has been introduced for the nonlinear cubic Duffing equation with and without damping effect. Since exact solutions of the corresponding equation for all initial guesses do not exist in the literature, we have first produced an exact solution for specific parameters by using the Kudryashov method to measure the accuracy of the currently suggested method. The innovative approach has been compared with the semi-analytic differential transform method and the fourth order RungeKutta method. Although the semi-analytic differential transform method is valid only for small time intervals, it has been proved that the innovative approach has ability to capture nonlinear behaviour of the process even in the long-time interval. In a comparison way, it has been shown that the present technique produces more accurate and computationally more economic results than the rival methods.

Drug resistance is a primary barrier to effective treatments of HIV/AIDS. Calculating quantitative relations between genotype and phenotype observations for each inhibitor with cell-based assays requires time and money-consuming experiments. Machine learning models are good options for tackling these problems by generalizing the available data with suitable linear or nonlinear mappings. The main aim of this study is to construct drug isolate fold (DIF) change-based artificial neural network (ANN) models for estimating the resistance potential of molecules inhibiting the HIV-1 protease (PR) enzyme. Throughout the study, seven of eight protease inhibitors (PIs) have been included in the training set and the remaining ones in the test set. We have obtained 11,803 genotype-phenotype data points for eight PIs from Stanford HIV drug resistance database. Using the leave-one-out (LVO) procedure, eight ANN models have been produced to measure the learning capacity of models from the descriptors of the inhibitors. Mean R2 value of eight ANN models for unseen inhibitors is 0.716, and the 95% confidence interval (CI) is [0.592–0.840]. Predicting the fold change resistance for hundreds of isolates allowed a robust comparison of drug pairs. These eight models have predicted the drug resistance tendencies of each inhibitor pair with the mean 2D correlation coefficient of 0.933 and 95% CI [0.930–0.938]. A classification problem has been created to predict the ordered relationship of the PIs, and the mean accuracy, sensitivity, specificity, and Matthews correlation coefficient (MCC) values are calculated as 0.954, 0.791, 0.791, and 0.688, respectively. Furthermore, we have created an external test dataset consisting of 51 unique known HIV-1 PR inhibitors and 87 genotype-phenotype relations. Our developed ANN model has accuracy and area under the curve (AUC) values of 0.749 and 0.818 to predict the ordered relationships of molecules on the same strain for the external dataset. The currently derived ANN models can accurately predict the drug resistance tendencies of PI pairs. This observation could help test new inhibitors with various isolates.

In this paper, numerical solutions of the advection-diffusion-reaction (ADR) equation are investigated using the Galerkin, collocation and Taylor-Galerkin cubic B-spline finite element method in strong form of spatial elements using an α-family optimization approach for time variation. The main objective of this article is to capture effective results of the finite element techniques with B-spline basis functions under the consideration of the ADR processes. All produced results are compared with the exact solution and the literature for various versions of problems including pure advection, pure diffusion, advection-diffusion, and advection-diffusion-reaction equations. It is proved that the present methods have good agreement with the exact solution and the literature.

In this study, an effective numerical method based on Taylor expansions is presented for boundary value problems. This method is arbitrary directional and called as implicit-explicit local differential transform method (IELDTM). With the completion of this study, a reliable numerical method is derived by optimizing the required degrees of freedom. It is shown that the order refinement procedure of the IELDTM does not affect the degrees of freedom. A priori error analysis of the current method is constructed and order conditions are presented in a detailed analysis. The theoretical order expectations are verified for nonlinear BVPs. Stability of the IELDTM is investigated by following the analysis of approximation matrices. To illustrate efficiency of the method, qualitative and quantitative results are presented for various challenging BVPs. It is tested that the current method is reliable and accurate for a broad range of problems even for strongly nonlinear BVPs. The produced results have revealed that the IELDTM is more accurate than the existing ones in literature.

In this study, the Burgers equation is analyzed both numerically and mathematically by considering various nite element based techniques including Galerkin, Taylor-Galerkin and collocation methods for spatial variation of the equation. The obtained time dependent ordinary di erential equation system is approximately solved by -family of time approximation. All these methods are theoretically explained using cubic B-spline basis and weight functions for a strong form of the model equation. Von Neumann matrix stability analysis is performed for each of these methods and stability criteria are determined in terms of the problem parameters. Some challenging examples of the Burgers equation are numerically solved and compared with the literature and exact solutions. Also, the proposed techniques have been compared with each other in terms of their advantages and disadvantages depending on the problem types. The more advantageous method of the three, in comparison to the other two, has been found for the special cases of the present problem in detail.