A multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues

Srivastava, Vineet K.; N., K.

doi:10.1016/j.mcm.2009.11.002

Cited by 82 publications

(37 citation statements)

References 16 publications

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“…Daftardar-Gejji and Jafari method (DJM) was first introduced by DaftardarGejji and Jafari [9] in 2006, it has been proved that this method is a better technique for solving different kinds of nonlinear equations [10,11,12,13,14,15,16,17,18,19,20]. DJM has been used to create a new predictor-corrector method [21,22].…”

Section: Resultsmentioning

confidence: 99%

Solving Strongly Nonlinear Oscillators by New Numerical Method

Batiha¹,

Ghanim²

2017

Int. J. of Pure and Appl. Math.

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Section: Resultsmentioning

confidence: 99%

Solving Strongly Nonlinear Oscillators by New Numerical Method

Batiha¹,

Ghanim²

2017

Int. J. of Pure and Appl. Math.

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“…In abnormal blood circulation, cells in the area most distal to the capillary at the venous end begin to suffer from hypoxia when perfusion levels drop to critically low values [1][2][3][4]. The insufficient oxygen delivery to tissue may interrupt metastasis and destroy part of the muscle.…”

Section: Introductionmentioning

confidence: 99%

“…Go [3] also identified that physical exertion boosts oxygen consumption rate but leads to depletion in oxygen from some cells and that cells engulfing capillaries can utilize or preserve more oxygen than required for a subsequent cycle. Srivastava and Rai [4] removed all the shortcomings discussed above and presented a new mathematical model for diffusion of oxygen of all types, i.e. for sub-diffusion and diffusion processes, using fractional calculus.…”

Section: Introductionmentioning

confidence: 99%

“…Analytical methods used to solve these equations have very restricted applications and the numerical techniques commonly used give rise to roundingoff errors. Keeping these in mind Srivastava and Rai [4] examined the effect of the concentration of the oxygen delivery from the capillary to the tissues by solving the linear and nonlinear multiterm fractional diffusion equations analytically using mathematical tools like the new iterative method (NIM) [6] and a modified Adomian decomposition method (MADM) [5].NIM was first proposed by the Indian mathematicians Daftardar-Gejji and Jafari [6] and subsequently implemented by Daftardar-Gejji and Bhalekar [7,8] to solve linear and nonlinear partial differential equations and fractional partial differential (diffusion wave) equations. The main advantage of this method is its computational simplicity; the solution obtained by this method is expected to give a better approximation in a straightforward manner.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, Schotet al [25] have proposed a generalized diffusion equation which employs space and time fractional derivatives, an absorbent (or source) term, and an external linear force field. This leads us for introducing the external forces along radial vector and axial line in our newly developed multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues [4].…”

Section: Introductionmentioning

confidence: 99%

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Numerical Study of Oxygen Diffusion From Capillary to Tissues During Hypoxia With External Force Effects

Srivastava

Tripathi

Bég

2017

J. Mech. Med. Biol.

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A mathematical model to simulate oxygen delivery through a capillary to tissues under the influence of an external force field is presented. The multi-term general fractional diffusion equation containing force terms and a time dependent absorbent term is taken into account.Fractional calculus is applied to describe the phenomenon of sub-diffusion of oxygen in both transverse and longitudinal directions. A new computational algorithm, i.e. the New Iterative Method (NIM) is employed to solve the spatio-temporal fractional partial differential equation subject to appropriate physical boundary conditions. Validation of NIM solutions is achieved with a Modified Adomian Decomposition Method (MADM). A parametric study is conducted for three loading scenarios on the capillary-radial force alone, axial force alone and the combined case of both forces. The results demonstrate that the force terms markedly influence the oxygen diffusion process. For example the radial force exerts a more profound effect than axial force on subdiffusion of oxygen indicating that careful manipulation of these forces on capillary-tissues may assist in the effective reduction of hypoxia or other oxygen depletion phenomena.

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A fast second‐order difference scheme for the space–time fractional equation

Wang

Sun

2019

Numerical Methods Partial

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In this paper, a fast second-order accurate difference scheme is proposed for solving the space-time fractional equation. The temporal Caputo derivative is approximated by  L2-1 formula which employs the sum-of-exponential approximation to the kernel function appeared in Caputo derivative. The second-order linear spline approximation is applied to the spatial Riemann-Liouville derivative. At each time step, a fast algorithm, the preconditioned conjugate gradient normal residual method with a circulant preconditioner (PCGNR), is used to solve the resulting system that reduces the storage and computational cost significantly. The unique solvability and unconditional convergence of the difference scheme are shown by the discrete energy method. Numerical examples are given to verify numerical accuracy and efficiency of the difference schemes. KEYWORDS convergence, fast method, finite difference scheme, space-time fractional equation, stability 1 Numer Methods Partial Differential Eq. 2019;35:1326-1342. wileyonlinelibrary.com/journal/num

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A multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues

Cited by 82 publications

References 16 publications

Solving Strongly Nonlinear Oscillators by New Numerical Method

Solving Strongly Nonlinear Oscillators by New Numerical Method

Numerical Study of Oxygen Diffusion From Capillary to Tissues During Hypoxia With External Force Effects

A fast second‐order difference scheme for the space–time fractional equation

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