Projects | Manoj Solanki

Projects

Compact Finite Difference Method to solve Differential Equations

Supervisor : Dr. Harshita Madduri

We solved various types of partial differential equations (PDEs), including linear and non-linear equations with constant and variable coefficients, such as the Black-Scholes equation and Burgers’ equation, using higher-order Compact Finite Difference Methods (CFDM). The implementation was carried out in Python, utilizing tools like Anaconda (Spyder), GPU computing, and Mathematica to enhance computational efficiency and accuracy.

Ongoing

Solution of Non Linear Partial Differential Equations

Supervisor : Dr. Harshita Madduri

We solved various types of differential equations, including the Burgers’ equation, using Newton’s Method, Quasi-Linearization Method, and the Finite Difference Method (FDM). The implementation was carried out in Python, leveraging numerical techniques to ensure accuracy and stability in the solutions.

Comparison of Solution of Parabolic PDEs

Supervisor : Dr. Harshita Madduri

We implemented the Heat Equation model using Explicit, Implicit, and Crank-Nicholson schemes and compared their computational costs. To solve the resulting Tri-Diagonal System of Linear Equations (SOLE) efficiently, we utilized the Thomas Algorithm and Sparse Matrix methods. The entire implementation was carried out using Python, focusing on accuracy, stability, and computational efficiency.

Package of Solution of ODE

Supervisor : Dr. R.P. Singh and Dr. Harshita Madduri

We have developed a comprehensive Python package for solving Ordinary Differential Equations (ODEs) using various numerical methods. This package efficiently handles both first-order and second-order ODEs, implementing methods such as Euler’s Method, Modified Euler’s Method (Heun’s Method), Runge-Kutta Methods (RK2, RK4), Taylor Series Method, Milne’s Predictor-Corrector Method, Adams-Bashforth Predictor-Corrector Method, and Runge-Kutta for Second-Order ODEs. Designed for students, researchers, and professionals in Applied Mathematics, Engineering, and Computational Science, it provides high accuracy and performance in solving differential equations. The package is ideal for mathematical modeling, engineering simulations, and computational science applications, where analytical solutions are impractical.

Codes for Numerical Methods in Python

Supervisor : Dr. R.P. Singh and Dr. Harshita Madduri

We developed a comprehensive Python codes for Numerical Methods, designed to solve various mathematical and engineering problems efficiently. These codes includes implementations of root-finding methods, numerical differentiation and integration, interpolation, linear and non-linear system solvers, and numerical solutions of ODEs and PDEs. It features multiple algorithms such as Newton’s Method, Secant Method, Runge-Kutta Methods The package is optimized for performance and accuracy, making it a valuable tool for students, researchers, and professionals in applied mathematics and computational science.

Codes for Finite Difference Methods

Supervisor : Prof. ASV Ravi Kanth

We have developed Python codes using Finite Difference Methods (FDM) to solve ODEs and PDEs efficiently. For parabolic PDEs, we implement FTCS, BTCS, and Crank-Nicholson methods for heat and diffusion problems. For hyperbolic PDEs, we use explicit schemes for wave propagation. For elliptic PDEs, we apply Jacobi, Gauss-Seidel, and SOR methods for steady-state solutions. Using Matplotlib, we visualize results through contour, surface, and time evolution plots, ensuring accuracy and efficiency for researchers and professionals.