A numerical approach for a class of time-fractional reaction–diffusion equation through exponential B-spline method

Kanth, A. S. V. Ravi; Garg, Neetu

doi:10.1007/s40314-019-1009-z

Cited by 21 publications

(21 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where [10,11]. Table 4 exhibits the maximum errors and order of convergence for ν = 0.6, different τ and h. Table 5 demonstrates that the comparison of computational and exact values corresponding different ν, N = 100 and τ = 0.005.…”

Section: Illustration Of Numerical Resultsmentioning

confidence: 97%

“…Example 1. Consider the TFRD of the form: [11]. Table 1 shows the comparison of computational and analytical values corresponding to various ν, N = 100 and τ = 0.005 at t = 1 2 .…”

Section: Illustration Of Numerical Resultsmentioning

confidence: 99%

“…Zeynab and Habibollah [29] solved the fractional reaction-convection-diffusion model numerically using wavelets operational matrices and B-spline scaling functions. Kanth and Garg [11] proposed the exponential cubic B-spline for solving the TFRD equation with Dirichlet boundary conditions. Pandey et al [30] obtained the numerical solution of TFRD equation in porous media using homotopy perturbation and Laplace transform.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Numerical Approach of a Time Fractional Reaction–Diffusion Model with a Non-Singular Kernel

et al. 2020

View full text Add to dashboard Cite

The time–fractional reaction–diffusion (TFRD) model has broad physical perspectives and theoretical interpretation, and its numerical techniques are of significant conceptual and applied importance. A numerical technique is constructed for the solution of the TFRD model with the non-singular kernel. The Caputo–Fabrizio operator is applied for the discretization of time levels while the extended cubic B-spline (ECBS) function is applied for the space direction. The ECBS function preserves geometrical invariability, convex hull and symmetry property. Unconditional stability and convergence analysis are also proved. The projected numerical method is tested on two numerical examples. The theoretical and numerical results demonstrate that the order of convergence of 2 in time and space directions.

show abstract

Section: Illustration Of Numerical Resultsmentioning

confidence: 97%

“…Example 1. Consider the TFRD of the form: [11]. Table 1 shows the comparison of computational and analytical values corresponding to various ν, N = 100 and τ = 0.005 at t = 1 2 .…”

Section: Illustration Of Numerical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Numerical Approach of a Time Fractional Reaction–Diffusion Model with a Non-Singular Kernel

et al. 2020

View full text Add to dashboard Cite

show abstract

“…Many processes in applied sciences and engineering can be modeled more accurately by fractional derivatives than integer order derivatives. In this paper, we consider the following time‐fractional reaction–diffusion (TFRD) equation [16]:

_{0} D_{t}^{α} u false(x, t false) = u_{italicxx} false(x, t false) - ν italicu false(x, t false) + f false(x, t false), false(x, t false) \in false[0, B_{u} false] \times false[0, T false], 0 < α < 1,

subject to the following initial condition (IC)

u false(x, 0 false) = μ false(x false),

and boundary conditions (BCs)

u false(0, t false) = θ_{1} false(t false), u false(B_{u}, t false) = θ_{2} false(t false),

where ν ≥ 0 and f ( x , t ), μ ( x ), θ 1 ( t ) and θ 2 ( t ) are sufficiently smooth functions. Further, the fractional derivative

_{0} D_{t}^{α} u false(x, t false)

in (1) is the Caputo derivative defined as

_{0} D_{t}^{α} u false(x, t false) = \frac{1}{Γ false(1 - α false)} \int_{0}^{t} {(t - ξ)}^{- α} \frac{\partial u false(x, ξ false)}{\partial ξ}

…”

Section: Introductionmentioning

confidence: 99%

A high order numerical scheme for solving a class of non‐homogeneous time‐fractional reaction diffusion equation

Roul

Goura

2020

Numerical Methods Partial

View full text Add to dashboard Cite

This paper is concerned with the development of a high order numerical technique for solving time‐fractional reaction–diffusion equation. The fractional derivative in the governing equation is described in the Caputo sense and a collocation method based on quintic B‐spline basis function is used to discretize the space variable. The stability and convergence analysis of the method are investigated, and it is shown that the proposed method converges to the exact solution of the problem with order of convergence O(Δt2 − α + Δx4), where α is the order of fractional derivative (0 < α < 1). Three test problems are considered to illustrate the efficiency and accuracy of present numerical scheme and to verify the theoretical result. It is shown that the order of fractional derivative has a profound effect on the solution of time‐fractional reaction–diffusion equation.

show abstract

“…Avazzadeh and Hassani (2019) applied transcendental Bernstein series as base functions to solve reaction-diffusion equations with nonlocal boundary conditions. Ravi Kanth and Garg (2020) presented a combination of the Crank-Nicolson and exponential B-spline methods for solving a class of time-fractional reaction-diffusion equation. Yang et al (2020) considered the Crank-Nicolson orthogonal spline collocation method for the approximate solution of the variable coefficient fractional mobile-immobile equation.…”

Section: Introductionmentioning

confidence: 99%

Solving nonlinear systems of fractional-order partial differential equations using an optimization technique based on generalized polynomials

et al. 2020

View full text Add to dashboard Cite

This paper addresses the application of generalized polynomials for solving nonlinear systems of fractional-order partial differential equations with initial conditions. First, the solutions are expanded by means of generalized polynomials through an operational matrix. The unknown free coefficients and control parameters of the expansion with generalized polynomials are evaluated by means of an optimization process relating the nonlinear systems of fractionalorder partial differential equations with the initial conditions. Then, the Lagrange multipliers are adopted for converting the problem into a system of algebraic equations. The convergence of the proposed method is analyzed. Several prototype problems show the applicability of the algorithm. The approximations obtained by other techniques are also tested confirming the high accuracy and computational efficiency of the proposed approach.

show abstract

A numerical approach for a class of time-fractional reaction–diffusion equation through exponential B-spline method

Cited by 21 publications

References 25 publications

A Numerical Approach of a Time Fractional Reaction–Diffusion Model with a Non-Singular Kernel

A Numerical Approach of a Time Fractional Reaction–Diffusion Model with a Non-Singular Kernel

A high order numerical scheme for solving a class of non‐homogeneous time‐fractional reaction diffusion equation

Solving nonlinear systems of fractional-order partial differential equations using an optimization technique based on generalized polynomials

Contact Info

Product

Resources

About