In this work, a numerical scheme based on combined Lucas and Fibonacci polynomials is proposed for one- and two-dimensional nonlinear advection–diffusion–reaction equations. Initially, the given partial differential equation (PDE) reduces to discrete form using finite difference method and $$\theta -$$ θ - weighted scheme. Thereafter, the unknown functions have been approximated by Lucas polynomial while their derivatives by Fibonacci polynomials. With the help of these approximations, the nonlinear PDE transforms into a system of algebraic equations which can be solved easily. Convergence of the method has been investigated theoretically as well as numerically. Performance of the proposed method has been verified with the help of some test problems. Efficiency of the technique is examined in terms of root mean square (RMS), $$L_2$$ L 2 and $$L_\infty $$ L ∞ error norms. The obtained results are then compared with those available in the literature.

This paper presents a numerical scheme based on Haar wavelet for the solutions of higher order linear and nonlinear boundary value problems. In nonlinear cases, quasilinearization has been applied to deal with nonlinearity. Then, through collocation approach computing solutions of boundary value problems reduces to solve a system of linear equations which are computationally easy. The performance of the proposed technique is portrayed on some linear and nonlinear test problems including tenth, twelfth, and thirteen orders. Further convergence of the proposed method is investigated via asymptotic expansion. Moreover, computed results have been matched with the existing results, which shows that our results are comparably better. It is observed from convergence theoretically and verified computationally that by increasing the resolution level the accuracy also increases.

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Fractional differential equations precisely measure and describe biological and physical processes because of the symmetry feature in nature. Multi-term time-fractional (MTTF) introduced for modeling of complex processes in different physical phenomena. This article introduces a numerical method based on Cubic B-spline (CBS) Finite element method (FEM) for solution of MTTF partial differential equations(PDEs). Finite difference method is combined with theta-weighted scheme and simple quadrature rules are used for time-fractional discretization while CBS FEM is employed for space approximation. In addition, four-point Gauss Legendre quadrature is applied to evaluate the source term. Stability of the proposed scheme is discussed by Von Neumann method which verifies that the scheme is unconditionally stable. L 2 and L ∞ norms are used to check efficiency and accuracy of the proposed scheme. Computed results are compared with the exact and available methods in literature which show betterment of the proposed method.

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