B. V. Rathish Kumar

Sunil Kumar, B. V. Rathish Kumar, J. H. M. Ten Thije Boonkkamp

In this study, we propose a new scheme named as complete flux scheme (CFS) based on the finite volume method for solving singularly perturbed differential-difference equations (SPDDEs) of elliptic type. An alternate integral representation for the flux is obtained which plays an important role in the derivation of CF scheme. We have established the stability, consistency and quadrature convergence of the proposed scheme. The scheme is successfully implemented on test problems.

Meena Pargaei, B. V. Rathish Kumar

In this work, we propose the Haar wavelet method for the coupled degenerate reaction-diffusion PDEs and the ODEs having non-linear a source with Neumann boundary, applicable in various fields of the natural sciences, engineering, and economics, for example in gas dynamics, certain biological models, assets pricing in economics, composite media etc. Convergence analysis of the proposed numerical scheme has been carried out. We use the GMRES solver to solve the linear system of equations. Numerical solutions for the model problems of medical significance have been successfully solved.

B. V. Rathish Kumar, Ayan Chakraborty

We propose a non uniform web spline based finite element analysis for elliptic partial differential equation with the gradient type nonlinearity in their principal coefficients like p-laplacian equation and Quasi-Newtonian fluid flow equations. We discuss the well-posednes of the problems and also derive the apriori error estimates for the proposed finite element analysis and obtain convergence rate of $\mathcal{O}(h^α)$ for $α> 0$.

Ayan Chakraborty, B. V. Rathish Kumar

We analyze the error of the WEB-S finite element method applied to elliptic systems with non-cooperative dominant coupling,with a mixed Dirichlet/Neumann/Robin boundary condition. This problem is strongly related to a posteriori error estimates, giving computable bounds for computational errors and detecting zones in the solution domain where such errors are too large and certain mesh refinements should be performed. These results are based on an extensive regularity analysis of the interface problems of concern.Finally, the error analysis is illustrated by numerical experiments.

Bishal Chhetri, B. V. Rathish Kumar

The dynamics of tumor-immune interactions within a complex tumor microenvironment are typically modeled using a system of ordinary differential equations or partial differential equations. These models introduce some unknown parameters that need to be estimated accurately and efficiently from the limited and noisy experimental data. Moreover, due to the intricate biological complexity and limitations in experimental measurements, tumor-immune dynamics are not fully understood, and therefore, only partial knowledge of the underlying physics may be available, resulting in unknown or missing terms within the system of equations. In this study, we develop a cancer biology-informed neural network model(CBINN) to infer the unknown parameters in the system of equations as well as to discover the missing physics from sparse and noisy measurements. We test the performance of the CBINN model on three distinct nonlinear compartmental tumor-immune models and evaluate its robustness across multiple synthetic noise levels. By harnessing these highly nonlinear dynamics, our CBINN framework effectively estimates the unknown model parameters and uncovers the underlying physical laws or mathematical structures that govern these biological systems, even from scattered and noisy measurements. The models chosen here represent the dynamic patterns commonly observed in compartmental models of tumor-immune interactions, thereby validating the generalizability and efficacy of our methodology.

Bishal Chhetri, B. V. Rathish Kumar

In this study, we present an interpretable deep learning framework for the early detection of breast cancer using quantitative features extracted from digitized fine needle aspirate (FNA) images of breast masses. Our deep neural network, using ReLU activations, the Adam optimizer, and a binary cross-entropy loss, delivers state-of-the-art classification performance, achieving an accuracy of 0.992, precision of 1.000, recall of 0.977, and an F1 score of 0.988. These results substantially exceed the benchmarks reported in the literature. We evaluated the model under identical protocols against a suite of well-established algorithms (logistic regression, decision trees, random forests, stochastic gradient descent, K-nearest neighbors, and XGBoost) and found the deep model consistently superior on the same metrics. Recognizing that high predictive accuracy alone is insufficient for clinical adoption due to the black-box nature of deep learning models, we incorporated model-agnostic Explainable AI techniques such as SHAP and LIME to produce feature-level attributions and human-readable visualizations. These explanations quantify the contribution of each feature to individual predictions, support error analysis, and increase clinician trust, thus bridging the gap between performance and interpretability for real-world clinical use. The concave points feature of the cell nuclei is found to be the most influential feature positively impacting the classification task. This insight can be very helpful in improving the diagnosis and treatment of breast cancer by highlighting the key characteristics of breast tumor.

Akash Anand, Ambuj Pandey, B. V. Rathish Kumar et al.

This text proposes a fast, rapidly convergent Nyström method for the solution of the Lippmann-Schwinger integral equation that mathematically models the scattering of time-harmonic acoustic waves by inhomogeneous obstacles, while allowing the material properties to jump across the interface. The method works with overlapping coordinate charts as a description of the given scatterer. In particular, it employs "partitions of unity" to simplify the implementation of high-order quadratures along with suitable changes of parametric variables to analytically resolve the singularities present in the integral operator to achieve desired accuracies in approximations. To deal with the discontinuous material interface in a high-order manner, a specialized quadrature is used in the boundary region. The approach further utilizes an FFT based strategy that uses equivalent source approximations to accelerate the evaluation of large number of interactions that arise in the approximation of the volumetric integral operator and thus achieves a reduced computational complexity of $O(N \log N)$ for an $N$-point discretization. A detailed discussion on the solution methodology along with a variety of numerical experiments to exemplify its performance in terms of both speed and accuracy are presented in this paper.