A family of iterative methods to solve nonlinear problems with applications in fractional differential equations

Abstract

In this work, first, a family of fourth‐order methods is proposed to solve nonlinear equations. The methods satisfy the Kung‐Traub optimality conjecture. By developing the methods into memory methods, their efficiency indices are increased. Then, the methods are extended to the multi‐step methods for finding the solutions to systems of problems. The formula for the order of convergence of the multi‐step iterative methods is , where is the step number of the methods. It is clear that computing the Jacobian matrix derivative evaluation and its inversion are expensive; therefore, we compute them only once in every cycle of the methods. The important feature of these multi‐step methods is their high‐efficiency index. Numerical examples that confirm the theoretical results are performed. In applications, some nonlinear problems related to the numerical approximation of fractional differential equations (FDEs) are constructed and solved by the proposed methods.

Solution of convection-diffusion model in groundwater pollution

“…In recent decades, many fractional devices have been developed, including thermal, mechanical, and electrical components 6 . There are many methods which are using for solving FPDEs, Including: radial basis functions(RBF) 7 , 8 , finite difference 9 – 15 , wavelets method 16 – 20 , spectral method 21 – 28 , local radial basis functions method 29 , 30 , finite element method 31 , Lagrange multiplier method 32 , iterative methods 33 , Crank-Nicolson method 34 . The fractional convection-diffusion equations(FCDEs) are a type of FPDEs that are widely used in various branches of science as computational mathematical models for simulations, such as: dissolved contaminant transport in groundwater, energy, and mass transformation, oil reservoir simulations, global weather, the diaspora of chemicals in reactors, etc.…”

Numerical Study of Time-Fractional Schrödinger Model in One-Dimensional Space Arising in Mathematical Physics

“…Numerous researchers have offered several powerful computational and analytical strategies for determining results for fractional-order differential problems, such as: differential transform scheme [19], new iterative strategy [20], Trial equation strategy [21], Adomian decomposition technique [22], generalized Taylor matrix method [23], Hermite collocation method [24], and many others [25][26][27]. The power series technique [28] is a common and straightforward scheme to find computational results for solving linear differential problems.…”