A numerical method for solving a fractional partial differential equation through converting it into an NLP problem

Ghandehari, Mohammad Ali Mohebbi; Ranjbar, Mojtaba

doi:10.1016/j.camwa.2013.01.003

Cited by 25 publications

(10 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In spite of using small values of N , we have highly accurate solutions by using the present method. Table presents the absolute errors at α = 0.5 with N = M = 4, θ = ϑ =− 0.5 and various choices of x and t , and a comparison with the results obtained in . This demonstrates that the numerical solutions are in good agreement with the exact solutions and that our results are more accurate than the results obtained from the methods presented in .…”

Section: Numerical Results and Comparisonssupporting

confidence: 56%

“…Furthermore, in Figure , the logarithmic graphs of MAEs ( l o g 10 E r r o r ) is displayed using the present algorithm at θ = ϑ =− 0.5 for α = 0.1,0.3,0.5,0.7,0.9 with various values of N ( N = M ). Clearly, the numerical errors decay exponentially as N increases.Example Consider the following linear inhomogeneous time‐fractional equation :

\frac{\partial^{α} u (x, t)}{\partial t^{α}} + \frac{∂u (x, t)}{∂x} - \frac{\partial^{2} u (x, t)}{\partial x^{2}} = \frac{2 t^{2 - α}}{Γ (2 - α)} + 2 x - 2,

subject to the boundary conditions

u (0, t) = t^{2}, u (1, t) = 1 + t^{2}

and the initial condition u ( x ,0) = x 2 . The exact solution of this problem is

u (x, t) = x^{2} + t^{2} .

…”

Section: Numerical Results and Comparisonsmentioning

confidence: 99%

“…Table II presents the absolute errors at˛D 0.5 with N D M D 4, Â D # D 0.5 and various choices of x and t, and a comparison with the results obtained in [48]. This demonstrates that the numerical solutions are in good agreement with the exact solutions and that our results are more accurate than the results obtained from the methods presented in [47,48]. The numerical results of Example 2 for different values of˛and are shown in Figure 2.…”

Section: Examplementioning

confidence: 96%

“…Ref. [48] . In spite of using small values of N, we have highly accurate solutions by using the present method.…”

Section: Examplementioning

confidence: 99%

See 3 more Smart Citations

A fractional‐order Jacobi Tau method for a class of time‐fractional PDEs with variable coefficients

Bhrawy

Zaky

2015

Math Methods in App Sciences

View full text Add to dashboard Cite

This paper presents a shifted fractional‐order Jacobi orthogonal function (SFJF) based on the definition of the classical Jacobi polynomial. A new fractional integral operational matrix of the SFJF is presented and derived. We propose the spectral Tau method, in conjunction with the operational matrices of the Riemann–Liouville fractional integral for SFJF and derivative for Jacobi polynomial, to solve a class of time‐fractional partial differential equations with variable coefficients. In this algorithm, the approximate solution is expanded by means of both SFJFs for temporal discretization and Jacobi polynomials for spatial discretization. The proposed tau scheme, both in temporal and spatial discretizations, successfully reduced such problem into a system of algebraic equations, which is far easier to be solved. Numerical results are provided to demonstrate the high accuracy and superiority of the proposed algorithm over existing ones. Copyright © 2015 John Wiley & Sons, Ltd.

show abstract

Section: Numerical Results and Comparisonssupporting

confidence: 56%

\frac{\partial^{α} u (x, t)}{\partial t^{α}} + \frac{∂u (x, t)}{∂x} - \frac{\partial^{2} u (x, t)}{\partial x^{2}} = \frac{2 t^{2 - α}}{Γ (2 - α)} + 2 x - 2,

subject to the boundary conditions

u (0, t) = t^{2}, u (1, t) = 1 + t^{2}

and the initial condition u ( x ,0) = x 2 . The exact solution of this problem is

u (x, t) = x^{2} + t^{2} .

…”

Section: Numerical Results and Comparisonsmentioning

confidence: 99%

Section: Examplementioning

confidence: 96%

“…Ref. [48] . In spite of using small values of N, we have highly accurate solutions by using the present method.…”

Section: Examplementioning

confidence: 99%

See 2 more Smart Citations

A fractional‐order Jacobi Tau method for a class of time‐fractional PDEs with variable coefficients

Bhrawy

Zaky

2015

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…It is widely and efficiently used to describe many phenomena arising in engineering, physics, economy and science. A family of numerical [1][2][3], semi-analytical [4][5], and analytical methods has been developed for solving ordinary and fractional differential equations [6][7]. Many partial differential equations of fractional order have been studied and solved.…”

Section: Introductionmentioning

confidence: 99%