Research Projects

Existing research, primarily focuses on one-dimensional models, highlighting the need to extend wavelet-based methods, such as Fibonacci and Taylor wavelets, to multidimensional fractional differential and integral equations to better address complex real-world phenomena. Additionally, while tumor growth model, malaria model, measles model, breast cancer models has been examined by using already existing techniques where there is a big issue of stability, so their analysis remains incomplete. Developing novel polynomial-based wavelet approaches could provide deeper insights into tumor dynamics and extend applications to other disease models, such as measles and breast cancer models. As polynomial-based wavelet methods have been applied to only few biological models, the development of versatile and efficient wavelet techniques tailored to complex nonlinear systems and biological applications remains an underexplored area. Followings are the objectives of this research proposal: (i) Propose different type of polynomial-based wavelets methods and undertake their convergence analysis. (ii) To apply the polynomial-based collocation wavelet methods to solve Stratonovich equations. (iii) Numerical analysis of linear and nonlinear systems of equations of biological models such as; tumor model, malaria model, measles model and breast cancer model etc by using proposed polynomial-based wavelets methods. (iv) To find the stability of the systems of equations, their convergence analysis and error estimation. (v) To solve the multidimensional fractional differential and integral equations by using different wavelets and compare the outcomes with the existing methods like RK4 method, Adomian decomposition method, Variational Iteration Method, Spectral method, Haar Wavelet method, B-spline method etc.The following methodology will be used  Mathematical modeling of tumor model, malaria model, measles model and breast cancer model and Stratonovich Integral Equations. With the help of mathematical modeling convert the physical problem into mathematical form. Construction of polynomial-based wavelet methods and corresponding function approximation. Caputo operator, Riemann-Liouville fractional integral operator, block plus function etc. will be used to calculate the fractional order derivatives. Corresponding Operational Matrix of integration(OMI), Operational Matrix of fractional integration(OMFI), Stochastic OMI, fractional order OMI. Solving the system of algebraic equations (linear/nonlinear) by using Newton method. Error estimation, convergence analysis and stability of the proposed methods. MATLAB/MATHEMATICA will be used to develop the codes to determine the unknown coefficients and plotting the graphs to understanding effects of fractional & Stochastic parameters. Get numerical results for different physical parameters, comparison of outcomes with the existing methods like RK4, Adomiandecomposition,Variational Iteration, Spectral, etc.