Multiple integral-balance method: Basic idea and an example with Mullin’s model of thermal grooving

A multiple integration technique of the integral-balance method allowing solving high-order diffusion equations is conceived in this note. The new method termed multiple-integral balance method is based on multiple integration procedures with respect to the space co-ordinate and is generalization of the widely applied heat-balance integral method of Goodman and the double integration method of Volkov. The method is demonstrated by a solution of the linear diffusion models of Mullins for thermal grooving.

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Fourth-order fractional diffusion model of thermal grooving: integral approach to approximate closed form solution of the Mullins model

Mathematical Modelling of Natural Phenomena ◽

10.1051/mmnp/2017080 ◽

2018 ◽

Vol 13 (1) ◽

pp. 6 ◽

Cited By ~ 7

Author(s):

Jordan Hristov

Keyword(s):

Integration Method ◽

Closed Form Solution ◽

Multiple Integral ◽

Fractional Diffusion ◽

Form Solution ◽

Balance Method ◽

Integral Approach ◽

Integration Technique ◽

Multiple Integration ◽

Space Coordinate

A multiple integration technique of the integral-balance method allowing solving high-order subdiffusion diffusion equations is presented in this article. The new method termed multiple-integral balance method (MIM) is based on multiple integration procedures with respect to the space coordinate. MIM is a generalization of the widely applied Heat-balance integral method of Goodman and the double integration method of Volkov. The method is demonstrated by a solution of the linear subdiffusion model of Mullins for thermal grooving by surface diffusion.

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An alternative integral-balance solutions to transient diffusion of heat (mass) by time-fractional semi-derivatives and semi-integrals: Fixed boundary conditions

Thermal Science ◽

10.2298/tsci150917010h ◽

2016 ◽

Vol 20 (6) ◽

pp. 1867-1878

Author(s):

Jordan Hristov

Keyword(s):

Heat Balance ◽

Neumann Boundary Condition ◽

Integral Method ◽

Integration Method ◽

Neumann Boundary ◽

New Approach ◽

Fixed Boundary ◽

Surface Gradient ◽

Approximation Errors ◽

Heat Balance Integral Method

A new approach to integral-balance solutions of the diffusion equation of heat (mass) with constant transport properties by applying time-fractional semi-derivatives and semi-integrals of Riemann-Liouville sense has been developed. The time-fractional semiderivatives and semiintegrals replace the surface gradient (temperature) which in the classical Heat-balance integral method (HBIM) of Goodman and the Double-integration method (DIM) should be expressed through the assumed profile. The application of semiderivatives and semiintegrals reduces the approximation errors to levels less than the ones exhibited by the classical HBIM and DIM. The method is exemplified by solutions of Dirichlet and Neumann boundary condition problems.

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A cubic heat balance integral method for one-dimensional melting of a finite thickness layer

International Journal of Heat and Mass Transfer ◽

10.1016/j.ijheatmasstransfer.2007.06.014 ◽

2007 ◽

Vol 50 (25-26) ◽

pp. 5305-5317 ◽

Cited By ~ 29

Author(s):

T.G. Myers ◽

S.L. Mitchell ◽

G. Muchatibaya ◽

M.Y. Myers

Keyword(s):

Heat Balance ◽

Integral Method ◽

Finite Thickness ◽

One Dimensional ◽

Heat Balance Integral Method ◽

Thickness Layer

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Fluvial thermal erosion: heat balance integral method

Earth Surface Processes and Landforms ◽

10.1002/esp.1489 ◽

2007 ◽

Vol 32 (12) ◽

pp. 1828-1840 ◽

Cited By ~ 22

Author(s):

R. Randriamazaoro ◽

L. Dupeyrat ◽

F. Costard ◽

E. Carey Gailhardis

Keyword(s):

Heat Balance ◽

Integral Method ◽

Thermal Erosion ◽

Heat Balance Integral Method

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Heat-balance integral to fractional (half-time) heat diffusion sub-model

Thermal Science ◽

10.2298/tsci1002291h ◽

2010 ◽

Vol 14 (2) ◽

pp. 291-316 ◽

Cited By ~ 29

Author(s):

Jordan Hristov

Keyword(s):

Diffusion Equation ◽

Heat Balance ◽

Integral Method ◽

Time Derivative ◽

Heat Diffusion ◽

Half Time ◽

Heat Diffusion Equation ◽

Parabolic Profile ◽

Heat Balance Integral Method ◽

The Heat Balance

The fractional (half-time) sub-model of the heat diffusion equation, known as Dirac-like evolution diffusion equation has been solved by the heat-balance integral method and a parabolic profile with unspecified exponent. The fractional heat-balance integral method has been tested with two classic examples: fixed temperature and fixed flux at the boundary. The heat-balance technique allows easily the convolution integral of the fractional half-time derivative to be solved as a convolution of the time-independent approximating function. The fractional sub-model provides an artificial boundary condition at the boundary that closes the set of the equations required to express all parameters of the approximating profile as function of the thermal layer depth. This allows the exponent of the parabolic profile to be defined by a straightforward manner. The elegant solution performed by the fractional heat-balance integral method has been analyzed and the main efforts have been oriented towards the evaluation of fractional (half-time) derivatives by use of approximate profile across the penetration layer.

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