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This paper is dedicated to proposing two numerical algorithms for solving the one- and two-dimensional heat partial differential equations (PDEs). In these algorithms, generalized Lucas polynomials (GLPs) involving two parameters are utilized as basis functions. The two proposed numerical schemes in one and two- dimensions are based on solving the corresponding integral equation to the heat equation, and after that employing, respectively, the tau and collocation methods to convert the heat equations subject to their underlying conditions into systems of linear algebraic equations that can be treated efficiently via suitable numerical procedures. In this article, the convergence analysis is examined for the proposed generalized Lucas expansion. Five illustrative problems are numerically solved via the two proposed numerical schemes to show the applicability and accuracy of the presented algorithms. Our obtained results compare favourably with the exact solutions. [ABSTRACT FROM AUTHOR]

In this paper, the existence and uniqueness of the Fredholm–Volterra integro-differential equation with the nonlocal condition will be studied. Also, we study the continuous dependence of the initial data. The numerical solution of the problem will be studied using the central difference approximations and trapezoidal rule to transform the Volterra–Fredholm integro-differential equation into a system of algebraic equations which can be solved together to get the solution. Finally, we solve some examples numerically to show the accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

This paper proposes a numerical method for solving fractional relaxation-oscillation equations. A relaxation oscillator is a type of oscillator that is based on how a physical system returns to equilibrium after being disrupted. The primary equation of relaxation and oscillation processes is the relaxation-oscillation equation. The fractional derivatives in the relaxation-oscillation equations under consideration are defined in the Φ -Caputo sense. The numerical method relies on a novel type of operational matrix method, namely, the Φ -Haar wavelet operational matrix method. The operational matrix approach has a lower computational complexity. The proposed scheme simplifies the main problem to a set of linear algebraic equations. Numerical examples demonstrate the validity and applicability of the proposed technique. [ABSTRACT FROM AUTHOR]

The objective of this paper is to present an efficient numerical technique for solving time fractional modified anomalous subdiffusion equation. Anomalous diffusion equation has its role in various branches of biological sciences. B-spline is a piecewise function to draw curves and surfaces, which maintain its degree of smoothness at the connecting points. B-spline provides an active process of approximation to the limit curve. In current attempt, B-spline curve is used to approximate the solution curve of time fractional modified anomalous subdiffusion equation. The process is kept simple involving collocation procedure to the data points. The time fractional derivative is approximated with the discretized form of the Riemann-Liouville derivative. The process results in the form of system of algebraic equations, which is solved using a variant of Thomas algorithm. In order to ensure the convergence of the procedure, a valid method named Von Neumann stability analysis is attempted. The graphical and tabular display of results for the illustrated examples is presented, which stamped the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The objective of this paper is to present an efficient numerical technique for solving time fractional modified anomalous subdiffusion equation. Anomalous diffusion equation has its role in various branches of biological sciences. B-spline is a piecewise function to draw curves and surfaces, which maintain its degree of smoothness at the connecting points. B-spline provides an active process of approximation to the limit curve. In current attempt, B-spline curve is used to approximate the solution curve of time fractional modified anomalous subdiffusion equation. The process is kept simple involving collocation procedure to the data points. The time fractional derivative is approximated with the discretized form of the Riemann-Liouville derivative. The process results in the form of system of algebraic equations, which is solved using a variant of Thomas algorithm. In order to ensure the convergence of the procedure, a valid method named Von Neumann stability analysis is attempted. The graphical and tabular display of results for the illustrated examples is presented, which stamped the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]