Muhammad Shoaib Arif^1,2,*, Muhammad Jhangir², Yasir Nawaz², Imran Abbas², Kamaleldin Abodayeh¹, Asad Ejaz²

CMES-Computer Modeling in Engineering & Sciences, Vol.133, No.2, pp. 303-325, 2022, DOI:10.32604/cmes.2022.020979 - 21 July 2022

Abstract The numerous applications of Maxwell Nanofluid Stagnation Point Flow, such as those in production industries, the processing of polymers, compression, power generation, lubrication systems, food manufacturing and air conditioning, among other applications, require further research into the effects of various parameters on flow phenomena. In this paper, a study has been carried out for the heat and mass transfer of Maxwell nanofluid flow over the heated stretching sheet. A mathematical model with constitutive expressions is constructed in partial differential equations (PDEs) through obligatory basic conservation laws. A series of transformations are then used to take… More >

Jadav Konch^*, G. C. Hazarika

Frontiers in Heat and Mass Transfer, Vol.8, pp. 1-10, 2017, DOI:10.5098/hmt.8.39

Abstract This paper investigates momentum, heat and mass transfer characteristics of a hydromagnetic Newtonian dusty fluid flow due to a rotating disk with radiation and viscous dissipation. The main objective of this paper is to study effects of temperature dependent viscosity and thermal conductivity on flow, temperature and species concentration. Radiation and viscous dissipation effects are also taken into account. Saffman model for dusty fluid is considered for the problem. The partial differential equations governing the flow are converted into ordinary differential equations employing similarity transformations. The resulting highly nonlinear coupled ordinary differential equations are solved More >

B. Ahmad^*, Z. Iqbal

Frontiers in Heat and Mass Transfer, Vol.8, pp. 1-6, 2017, DOI:10.5098/hmt.8.22

Abstract We examine the behavior of Cattaneo-Christov heat flux model for two-dimensional incompressible flow of Eyring Powell fluid passed over an exponentially stretching sheet. Mathematical formulation is performed by assuming boundary layer approximation. Cattaneo Christov heat flux model is applied to analyze the heat transport phenomenon. Thermal relaxation time is envisaged on the layer induced due to boundary. The governing Partial Differential equations are converted into Ordinary differential equations by the appropriate use of similarity transformation. Shooting approach is used to tackle the obtained boundary layer equations. The effects of obtained similarity parameters are plotted and More >

S. Abbasbandy^1,2, R.A. Van Gorder³, M. Hajiketabi¹

CMES-Computer Modeling in Engineering & Sciences, Vol.104, No.4, pp. 329-351, 2015, DOI:10.3970/cmes.2015.104.329

Abstract We transform the Yamabe equation on a ball of arbitrary dimension greater than two into a nonlinear singularly boundary value problem on the unit interval [0,1]. Then we apply Lie-group shooting method (LGSM) to search a missing initial condition of slope through a weighting factor r ∈ (0,1). The best r is determined by matching the right-end boundary condition. When the initial slope is available we can apply the group preserving scheme (GPS) to calculate the solution, which is highly accurate. By LGSM we obtain precise radial symmetric solutions of the Yamabe equation. These results are useful More >

Chein-Shan Liu^1,2, Chung-Lun Kuo³, Dongjie Liu⁴

CMC-Computers, Materials & Continua, Vol.24, No.2, pp. 105-124, 2011, DOI:10.3970/cmc.2011.024.105

Abstract The inverse Cauchy problems for elliptic equations, such as the Laplace equation, the Poisson equation, the Helmholtz equation and the modified Helmholtz equation, defined in annular domains are investigated. The outer boundary of the annulus is imposed by overspecified boundary data, and we seek unknown data on the inner boundary through the numerical solution by a spring-damping regularization method and its Lie-group shooting method (LGSM). Several numerical examples are examined to show that the LGSM can overcome the ill-posed behavior of inverse Cauchy problem against the disturbance from random noise, and the computational cost is More >

Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.56, No.1, pp. 85-112, 2010, DOI:10.3970/cmes.2010.056.085

Abstract We propose a novel technique, transforming the generalized SturmLiouville problem: w'' + q(x,λ)w = 0, a₁(λ)w(0) + a₂(λ)w'(0) = 0, b₁(λ)w(1) + b₂(λ)w'(1) = 0 into a canonical one: y'' = f, y(0) = y(1) = c(λ). Then we can construct a very effective Lie-group shooting method (LGSM) to compute eigenvalues and eigenfunctions, since both the left-boundary conditions y(0) = c(λ) and y'(0) = A(λ) can be expressed explicitly in terms of the eigen-parameter λ. Hence, the eigenvalues and eigenfunctions can be easily calculated with better accuracy, by a finer adjusting of λ to match the right-boundary condition y(1) = c(λ). Numerical examples More >

Weichung Yeih^1,2, Chein-Shan Liu³

CMES-Computer Modeling in Engineering & Sciences, Vol.46, No.2, pp. 107-128, 2009, DOI:10.3970/cmes.2009.046.107

Abstract We consider an inverse problem for estimating an unknown time dependent heat source H(t) in a heat conduction equation u_t(x,t) = u_xx(x,t) + H(t). First this inverse problem is formulated as a three-point boundary value problem (BVP) for ODEs discretized from the transformed homogeneous governing equation. To treat this three-point BVP we develop a two-stage Lie-group shooting method (TSLGSM). The novel approach is examined through numerical examples to convince that it is rather accurate and efficient; the estimation error is small even for identifying discontinuous and oscillatory heat sources under noise. More >

Chein-Shan Liu¹

The International Conference on Computational & Experimental Engineering and Sciences, Vol.5, No.2, pp. 55-84, 2008, DOI:10.3970/icces.2008.005.055

Abstract The present paper provides a Lie-group shooting method for the numerical solutions of second-order nonlinear boundary value problems exhibiting multiple solutions. It aims to find all solutions as easy as possible. The boundary conditions considered are classified into four types, namely the Dirichlet, the first Robin, the second Robin and the Neumann. The two Robin type problems are transformed into a canonical one by using the technique of symmetric extension of the governing equations. The Lie-group shooting method is very effective to search unknown initial condition through a weighting factor r(0,1). Furthermore, the closed-form solutions are More >

Chein-Shan Liu¹, Satya N. Atluri²

CMES-Computer Modeling in Engineering & Sciences, Vol.36, No.3, pp. 261-286, 2008, DOI:10.3970/cmes.2008.036.261

Abstract The inverse Sturm-Liouville problem finds its applications in the identification of mechanical properties and/or geometrical configurations of a vibrating continuous medium; however, this problem is hard to solve, either theoretically or numerically. Previously, Liu (2008a) has constructed a Lie-group shooting method to determine the eigenvalues, and the corresponding eigenfunctions, for the direct Sturm-Liouville problem. In this study, we are concerned with solving the inverse Sturm-Liouville problem, by developing a Lie-group of SL(2,R) to construct nonlinear algebraic equations (NAEs), when discrete eigenvalues are specified. Our purpose here is to use these NAEs to solve the unknown function More >

Chein-Shan Liu ¹

CMES-Computer Modeling in Engineering & Sciences, Vol.35, No.2, pp. 157-180, 2008, DOI:10.3970/cmes.2008.035.157

Abstract For an inverse vibration problem of nonlinear mechanical system to estimate displacement- and velocity-dependent restoring force, we transform the equation of motion into a parabolic type partial differential equation (PDE). Then by a semi-discretization of the PDE, the inverse vibration problem is formulated as a multi-dimensional two-point boundary value problem with unknown sources, allowing a closed-form estimation through a Lie-group shooting method to construct the restoring force surface over phase plane. Only one set of displacements measured at sampling time points is used in the estimation. The new method does not require to assume a More >