Startseite Chebyshev Operational Matrix Method for Lane-Emden Problem
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Chebyshev Operational Matrix Method for Lane-Emden Problem

  • Bhuvnesh Sharma , Sunil Kumar EMAIL logo , M.K. Paswan und Dindayal Mahato
Veröffentlicht/Copyright: 22. Mai 2018
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Nonlinear Engineering
Aus der Zeitschrift Nonlinear Engineering Band 8 Heft 1

Abstract

In the this paper, a new modified method is proposed for solving linear and nonlinear Lane-Emden type equations using first kind Chebyshev operational matrix of differentiation. The properties of first kind Chebyshev polynomial and their shifted polynomial are first presented. These properties together with the operation matrix of differentiation of first kind Chebyshev polynomial are utilized to obtain numerical solutions of a class of linear and nonlinear LaneEmden type singular initial value problems (IVPs). The absolute error of this method is graphically presented. The proposed framework is different from other numerical methods and can be used in differential equations of the same type. Several examples are illuminated to reveal the accuracy and validity of the proposed method.

Keywords: Lane-Emden equations; chebyshev first kind operational matrix of differentiation; Isothermal gas spheres equation; Nonlinear ODE

1 Introduction

In astrophysics, Lane-Emden equation model is one of the case of singular initial value problem in the form of second-order nonlinear ordinary differential equation(ODE), which describes several important phenomenon in mathematical science and astrophysics such as radiative cooling, behavior of spherical self-gravitating gas clouds, stellar structure [1], thermal explosions [2], modeling of cluster of galaxies, treatment of a phase transition in critical absorption, thermionic currents [3]. The dimensionless Poisson’s equation which describes the equilibrium density distribution in self-gravitating symmetric sphere of polytropic isothermal gas, was proposed by Jonathan H. & Lane [4]. The further equation described by Robert & Emden [5] which is known as Lane-Emden equation. This equation dictates the pressure and density variation with respect to each other, formulated as

y″(x)+αxy′(x)+f(x,y)=g(x)α,x≥0,(1)

subject to supplementary conditions:

y(0)=A,y′(0)=0(2)

where A is constant, f(x, y) is a continuous real valued function, and g(x) ∈C[0, 1]. It is known that the analytical solution of lane-Emden equation exists for supplementary conditions(2) in the locality of the singular point at x = 0. Taking α = 2, g(x) = 0, f(x, y) = yn and A = 1 in equation (1) and (2) respectively

y″(x)+2xy′(x)+yn=0(3)

characteristically, Eq. (3), represents the standard Lane-Emden equation, which will be subjected to the supplementary conditions as given in Eq. (4).

y(0)=1,y′(0)=0(4)

Similarly, Isothermal gas spheres are modeled by Davis [6]

y″(x)+2xy′(x)+ey=0(5)

subject to supplementary conditions:

y(0)=0,y′(0)=0(6)

Various methods have been presented to solve the Lane-Emden equation in unbound domain. Wazwaz et al. [7] presented the series solution of integral form of Lane-Emden equation by using Adomian Decomposition method. Bender et al. [8] proposed a new perturbative technique by replacing nonlinear terms in the Lagrangian and introduced a new parameter ?. Aminikhah et al. [9] presented the numerical solution of Lane-Emden equation based on cubic B-spline approximation. Krivec and Mandelzweig [10, 11] proposed quasilinearization approach to solve the Lane-Emden equation with singular potentials. Ramos [12] introduced the linearization methods for the piece-wise closed form solution of Lane-Emden equation, which depends on the Jacobian and derivative of independent variable. The other effective methods are variation iterative methods [13], non-perturbative approximation method [14] and the Pade-series method [15]. Russell and Shampine [16] explored the Lane-Emden equation for linear function f(x, y) = ky + h(x) and proved that a unique solution exists if h(x) ∈ C[0, 1] and -∞ < k ≤ π2.

Hermite, Laguerre, Legendre and Chebyshev polynomials are orthogonal over unbounded domain. These polynomial spectral methods are most common methods due to reveal the more accurate solution of a class of Lane-Emden equation type [17, 18, 19, 20]. Parand et al. [21] proposed Hermite collocation method for Lane-Emden equation. Pandey et al. [22] developed efficient numerical method for solving Lane-Emden equation using Legendre operation matrix of differentiation. Doha et al. [23] presented a new algorithm using second kind chebyshev operational matrix of derivative to obtain numerical solutions of Lane-Emden type IVPs. Abd-Elhameed et al. [24, 25] proposed a new algorithm to describe Lane-Emden Singular type IVPs using new Galerkin operation matrix of differentiation.

Wavelet theory is a relatively new and comes into view of mathematical research area. It has been applied with the orthogonal polynomials to propose a new method for solving several engineering problems. Yousefi [26] has given the solution of Lane-Emden equation using Legendre wavelet method. Youssri et al. [27] introduced ultraspherical wavelets operational matrix of derivatives for approximation of a class of Lane-Emden type with high accuracy.

The aim of this paper is to develop a more accurate and fast computational method for solving linear and nonlinear Lane-Emden equation (1) using first kind Chebyshev operation matrix of differentiation. The manuscript is organized as follows: Section 2 introduces the definition of first kind Chebyshev polynomial and formulates differential operation matrix. Section 3 describes the application of differentiation matrix of Chebyshev polynomial for solving the Lane-Emden equation. Section 4 exhibits the approximate solution obtained by the proposed method for the case of physical significance of polytropic index range 0 < n ≤ 5, wherein the result shows high accuracy and efficiency than other existing numerical solutions.

2 First kind chebyshev polynomials and its operational matrix of Differentiation

The analytic form of the shifted chebyshev polynomial Pm(x) of degree m are given by Atabakzadeh et al. [28]

Pm(x)=∑k=0[m2](−1)k2m−2k−1mm−km−kk(2x−1)m−2k(7)

Any function, y(x) ∈ L2[0, 1] can be approximated as a sum of shifted chebyshev polynomial (of the first kind) as:

y(x)=∑i=0∞ciPi(x)(8)

where

ci=〈y(x),Pi(x)〉=(2i+1)∫01y(x)Pi(x)dx,i=0,1,⋯(9)

In general, the series in Equation(8) can be truncated with the first (N + 1) shifted chebyshev polynomial of first kind as

yN(x)=∑i=0NciPi(x)=CTϕ(x)(10)

where

CT=[c0c1⋯cN],ϕ(x)=[P0(x)P1(x)⋯PN(x)]T

The operation matrix of derivative of the shifted chebyshev polynomial set ϕ(x) is defined as

dϕ(x)dx=D(1)ϕ(x)(11)

where D(1) in the (N + 1) × (N + 1) operation matrix of differentiation of chebyshev polynomial given as:

D(1)=dij=4iηj,j=i−kk=1,3,...,Nif;Neven,k=1,3,...,N−1ifNodd,0otherwise,(12)

where η0 = 2 and ηk = 1 (k ≥ 1).

For example N = 6, we have

D(1)=200000010000004000030600008080050100100(13)

where m = 0, 1, 2 ⋯ N.

From Eq. (10), it can be generalized for any n ∈ N as:

dnϕ(x)dx=(D(1))nϕ(x)=Dnϕ(x),wheren=1,2,3,⋯(14)

3 Application of the method

This section describe the application of differentiation matrix of chebyshev polynomial for solving the Lane-Emden equation. The Lane-Emden equation is formulated as

y″(x)+αxy′(x)+f(x,y)=g(x)α,x≥0,

subject to supplementary conditions: y(0) = A, y′(0) = 0. Approximating y(x), f(x, y) and g(x) by the shifted chebyshev polynomial has

y(x)≈∑i=0nciPi(x)=CTϕ(x),(15)
f(x,y)≈f(x,CTϕ(x)),(16)
g(x)≈∑i=0ngiPi(x)=GTϕ(x),(17)

where the unknowns are, C = [c0c1 ⋯ cN]T.

Using first kind chebyshev polynomial operation matrix of derivative, equation (15) can be expressed as

CTD2ϕ(x)+αxCTD1ϕ(x)+f(x,CTϕ(x))≈GTϕ(x)(18)

The Residual RN(x) for equation(18) can be expressed as

RN(x)≈CTD2ϕ(x)+αxCTD1ϕ(x)+f(x,CTϕ(x))−GTϕ(x)(19)

Eq. (18) is converted in N − 2 nonlinear equations by applying Tau method [29], using following equation,

〈RN(x),Pi(x)〉=∫01RN(x)Pi(x)dx=0,i=0,1,2,⋯,N−2.(20)

The supplementary conditions of Eq. (17) are given by

y(0)=CTϕ(0)=d0,y(1)(0)=CTD(1)ϕ(0)=d1.(21)

By substitute the shifted polynomials in Eq. (20), we have (N − 2) nonlinear equations. These equation together with Eq. (24) generate N nonlinear equations. This nonlinear system can be solved using Newton iterative method to determine coefficients of vector C. We can get the numerical solution y(x) of Lane-Emden equation by substituting C into Eq. (15).

4 Convergence and error analysis

In this section a convergence analysis is given for the method of solution discussed in last section for linear system of differential equations. It is well-known that shifted Chebyshev polynomials τi(x) forms a complete orthogonal set [30], Ω = [0, 1] and Lω2(Ω) denotes the space of all functions u : Ω → ℝ with weighted L2-norm and defined by

||u||Lω2(Ω)=∫01u2(t)ω(t)dt

we recall that Hω(Ω) is the sobolev space of all function u(t) on Ω such that u(t) and all its weak derivative up to order m are in Lω and define ||⋅||Hωl,N(Ω) as

||u||Hωm(Ω)=(∑k=0m||∂k∂tku(t)||Lω2(Ω)2)12

The semi-norm also defined by

|u|Hωm,N(Ω)=∑i=min(m,N)NN||u(i)(t)||Lω2(Ω)2

Now, suppose uN=∑k=0NU^kτk is the truncated Chebyshev approximations of a function u∈Hωm(Ω). Then, as has been proved in [30], the truncation error is

||u−uN||Hωl,N(Ω)≤CN2l−l/2−m|u|Hωm,N(Ω)(22)

For any 1 ≤ l ≤ m and C is a positive constant independent of m. Then, using proposed analytical approximation method, the approximation of y is yN = (y1N,y1N,...,ynN),yiN(x)=∑k=0Ncikτk(x) and error is defined by e(x) = (e1, e2, …, en) = y(x) = yN(x). Therefore,

D2ei+∑j=1kiaijej=egi,i=1,2,...,n,(23)

with egi = gi−giN and gikτk(x). Taking the norm ||⋅||Hωl,N(Ω) from both sides of (23) yields

||egi||Hωl,N(Ω)=||D2ei+∑j=1kiai,jej||Hωl,N(Ω)≤||D2ei||+∑j=1ki|ai,j|||ej||Hωl,N(Ω).(24)

On the other hand, since operator D2 on Chebyshev polynomial space is continuous and bounded, as has been proved in [31], there exists a constant C such that ||D2ei||Hωl,N(Ω)≤C¯||ei||Hωl,N(Ω). Now, using this and Eq. (24) we find

||egi||Hωl,N(Ω)≤C¯||ei||Hωl,N(Ω)+∑j=1ki|ai,j|||ej||Hωl,N(Ω).(25)

Let the vector norm ||e||Hωl,N(Ω)=max1≤i≤n||ei||Hωl,N(Ω) and the vector semi-norm |y|Hωl,N(Ω)max1≤i≤n||yi||Hωl,N(Ω). Then Eq. (25) yields

||egi||Hωl,N(Ω)≤C¯||e||Hωl,N(Ω)+∑j=1ki|ai,j|||e||Hωl,N(Ω)≤||e||Hωl,N(Ω)(C¯+∑j=1ki|ai,j|).(26)

Finally, combining (22) and (26) gives the following bound for the error in our presented analytical approximation method

||egi||Hωl,N(Ω)A≤||e||Hωl,N(Ω)≤CN2l−l/2−m|y|Hωm,N(Ω)

where A=C¯+max1≤i≤n∑j=1ki|ai,j|. Note that single Ω is compact set, |y|Hωm,N(Ω) is bounded and this means that our approximation converges for sufficiently large N.

Remark 1

As we have stated above, the convergence analysis for the proposed method just applied into the linear system (20). However, convergence analysis cannot simply determine for the nonlinear case as above discussion and needs some more efforts. Indeed for each nonlinear case, we may need some additional conditions on nonlinear functions yn(x) in system (20).

5 Numerical results and discussions

In this section, the introduced first kind Chebyshev operational matrix of derivatives is employed for solving both linear ana nonlinear Lane-Emden IVPs. The singularity in the variable coefficient (α/x) is handled by reducing a class of Lane-Emden type singular initial value problems to a system of algebraic equations using Tau method and supplementary conditions. The Newton iterative method is used to determine the coefficients of system equations. Six examples are executed on MATLAB-2016 using the proposed numerical algorithm to show the accuracy and efficiency of Chebyshev operation matrix method.

Let f(x, y) = yn(x), g(x) = 0 and A = 1, Eq.(1) can be written as

y″(x)+2xy′(x)+yn(x)=0(27)

with supplementary conditions

y(0)=1,y′(0)=0(28)

Example 1

for n = 0, the linear Lane-Emden equation from Eq. (27) has the form

y″(x)+2xy′(x)+1=0(29)

with supplementary conditions

y(0)=1,y′(0)=0(30)

Applying the method, proposed in section 3 for N = 2, we have

y(x)=CTϕ(x)=c0P0(x)+c1P1(x)+c2P2(x)(31)

From Eq.(18), D(1)=000200080andD(2)=0000001200.

By using Eq. (25), we obtain

48c2+1=0,(32)

By applying the supplementary conditions from Eq. (27) we have

c0−c1+c2=1,(33)
2c1−8c2=0,(34)

Solving Eqs. (32)-(34) we get,

c0=1516,c1=−112,c2=−148.

Substituting the values of constant vector in Eq. (31), the solution is obtained as:

y(x)=c0P0(x)+c1P1(x)+c2P2(x)=1516−112−14812x−18x2−8x+1=1−x26,

which is the exact solution.

Example 2

For n = 1, the nonlinear Lane-Emden equation from Eq. (27) has the form

y″(x)+2xy′(x)+y(x)=0(35)

with supplementary conditions

y(0)=1,   y′(0)=0(36)

Solving Eq. (35), the exact solution is y(x)=sinxx.

Here,

f(x,y)=y(x)=CTϕ(x).

The proposed method has been applied into the Eq. (35) with supplementary conditions Eq. (36) and the unknown matrix CT has evaluated by solving the nonlinear system for N = 8 and N = 10. The values of coefficients are are shown in Table 4 and 5 respectively.The exact solution and obtained solution using shifted first kind chebyshev polynomial in larger range [0, 8] are plotted in Fig. 1 and absolute error is graphically presented in Fig. 2.

Fig. 1 Plot between x & obtained solution y(x) for N = 8, N = 10 and exact solution.
Fig. 1

Plot between x & obtained solution y(x) for N = 8, N = 10 and exact solution.

Fig. 2 Absolute errors at N = 8 and N = 10 for example 2.
Fig. 2

Absolute errors at N = 8 and N = 10 for example 2.

Example 3

For n = 3, the nonlinear Lane-Emden equation from Eq. (27) has the form

y″(x)+2xy′(x)+y3(x)=0,(37)

with supplementary conditions

y(0)=1,   y′(0)=0(38)

Here,

y(x)=CTϕ(x)    and    f(x,y)=y3(x)=(CTϕ(x))3.

The values of unknown coefficients CT for N = 8 and N = 10 are shown in table 4 and 5 respectively. The analytic series solution [7] and obtained result using first kind chebyshev polynomial are graphically presented in Fig. 3 in range [0, 5].

Fig. 3 Plot of obtained solution y(x) for N = 8, N = 10 and series solution presented by Wazwaz [7] for example 3.
Fig. 3

Plot of obtained solution y(x) for N = 8, N = 10 and series solution presented by Wazwaz [7] for example 3.

The Table 1 shows the comparison of results and absolute errors with Wazwaz et al. [7] and Pandey et al. [22]. From Table 1 and Fig. 3, we observe that the proposed method providing better results compare to the recently developed methods.

Table 1

Comparison of approximate solutions and error obtained by modified method with series solution wazwaz [7] and newly advanced method Pandey [22] for Example 3.

xPresent methodWazwaz [7]Error at N =10Error at Pandey [22]
00.999999999912.23E-121.14E-13
0.10.99833582950.99833582952.12E-111.86E-11
0.20.99337309350.99337309351.38E-111.65E-11
0.50.95983906990.95983911444.45E-084.45E-08
10.85505756850.85509505433.74E-053.74E-05

Example 4

For n = 5, the nonlinear Lane-Emden equation from Eq. (27) has the form

y″(x)+2xy′(x)+y5(x)=0,(39)

with supplementary conditions

y(0)=1,   y′(0)=0(40)

having exact solution y(x)=(1+x23)−12.

Here,

y(x)=CTϕ(x)

and

f(x,y)=y5(x)=(CTϕ(x))5.

The unknown matrix CT has evaluated by solving the nonlinear system for N = 8 and N = 10. The values of coefficients are are shown in Table 4 and 5 respectively.The exact solution and obtained solution using shifted first kind chebyshev polynomial in larger range [0,5] are plotted in Fig. 4 and absolute error is graphically presented in Fig. 5.

Fig. 4 Plot of obtained solution y(x) for N = 8, N = 10 and exact solution for example 4.
Fig. 4

Plot of obtained solution y(x) for N = 8, N = 10 and exact solution for example 4.

Fig. 5 Absolute errors at N = 8 and N = 10 for example 4.
Fig. 5

Absolute errors at N = 8 and N = 10 for example 4.

Example 5

Isothermal gas spheres equation from Eq.(6) has the form

y″(x)+2xy′(x)+exp(y)=0,        x≥0,(41)

with supplementary conditions

y(0)=0,   y′(0)=0(42)

Here, we have

y(x)=CTϕ(x)   and   f(x,y)=ey.

Expansion of f(x, y) according to Taylor series is

f(x,y)=ey=1+y+y22+y36+y424+y5120+⋯(43)

By considering first five terms, f(x, y) can be expressed as

f(x,y)=1+CTϕ(x)+(CTϕ(x))22+(CTϕ(x))36+(CTϕ(x))424(44)

The unknown matrix CT is shown in Tables 4 and 5 respectively. The Wazwaz [7] analytic series solution and evaluated results by solving nonlinear system using developed method for N = 8 and N = 10, are shown in Fig. 6 in range [0,1].

Fig. 6 Plot of obtained solution y(x) for N = 8, N = 10 and series solution presented by Wazwaz [7] for example 5.
Fig. 6

Plot of obtained solution y(x) for N = 8, N = 10 and series solution presented by Wazwaz [7] for example 5.

The Table 2 shows the superiority of obtained approximate solutions than Wazwaz et al. [7], Pandey et al. [22] and Parand et al. [21].

Table 2

Comparison of approximate solutions and error obtained by modified method with series solution wazwaz [7], newly advanced method Pandey [22] and Parand [21] for Example 5.

xPresent methodWazwaz [7]Error for N =10Error at [22]Error at [21]
09.411877E-110.00000000009.41E-119.24E-180.00E+00
0.1-0.0016658337-0.00166583389.93E-115.28 E-105.85E-07
0.2-0.0066533669-0.00665336711.06E-103.37E-086.04E-07
0.5-0.0411539572-0.04115395673.42E-108.12E-065.58E-07
1-0.1588276829-0.15882728573.29E-074.93E-048.20E-07

Example 6

Consider the ordinary differential equation of Lane-Emden type of shape factor 2,

y″(x)+2xy′(x)+sin(y)=0,(45)

with supplementary conditions

y(0)=1,   y′(0)=0(46)

Here,

y(x)=CTϕ(x)  and  f(x,y)=sin(y).

Expansion of f(x, y) according to Taylor series is

f(x,y)=sin(y)=y−y36+y5120+⋯(47)

By considering only first three terms, f(x, y) can be expressed as

f(x,y)=CTϕ(x)−(CTϕ(x))36+(CTϕ(x))5120(48)

The coefficient vector CT is solved for N = 8 & N = 10 and shown in Tables 4 and 5 respectively. The obtained results are presented in Fig. 7 in range [0,1]. Table 3 presents the comparison of solution y(x) and absolute error with Wazwaz et al. [7], Pandey et al. [22] and Saadatmandi and Dehghan [29].

Fig. 7 Plot of obtained solution y(x) for N = 8, N = 10 and series solution presented by Wazwaz [7] for example 6.
Fig. 7

Plot of obtained solution y(x) for N = 8, N = 10 and series solution presented by Wazwaz [7] for example 6.

Table 3

Comparison of approximate solutions and error obtained by modified method with series solution wazwaz [7], and newly advanced method Pandey [22] & Saadtmandi and Dehgan [29] for example 6.

xPresent nethodWazwaz [7]Error at N=10Error at [22]Error at [29]
00.99999999991.00000000003.78E-141.11E-160.00E+00
0.10.99859760220.99859792735.83E-083.25E-077.21E-06
0.20.99439497690.99439626485.42E-071.28E-061.00E-05
0.50.96517024870.96517777971.67E-077.53E-061.04E-05
10.86365716760.86368073153.27E-062.35E-057.03E-06

Table 4

The value of the unknowns vector CT for discussed examples for N = 8.

coefficientsExample 2Example 3Example 4Example 5Example 6
c00.93973440.94358390.9460855-0.06033380.9484656
c1-0.0797633-0.0736793-0.0686545-0.0798799-0.0684238
c2-0.0190577-0.0161548-0.0138788-0.0191298-0.0166709
c30.00049990.00122240.00169770.00046860.0002518
c40.00005880.00010010.00008530.00004990.0000338
c5-0.0000012-0.0000145-0.0000309-2.536E-076.782E-07
c68.702E-08-3.823E-077.421E-07-1.444E-073.324E-08
c71.289E-091.398E-074.318E-071.293E-08-5.028E-09
c87.557E-119.082E-103.783E-083.846E-10-2.986E-10

Table 5

The value of the unknowns vector CT for discussed examples for N = 10.

coefficientsExample 2Example 3Example 4Example 5Example 6
c00.93973440.94358390.9468055-0.06033380.9484656
c1-0.0797633-0.0736793-0.0686545-0.0798799-0.0684238
c2-0.0190577-0.0161548-0.0138788-0.0191298-0.0166709
c30.00049990.00122240.00169770.00046860.0002518
c40.00005880.00010010.00008530.00004990.0000338
c5-0.0000012-0.0000145-0.0000309-2.536E-076.782E-07
c68.702E-08-3.812E-074.794E-07-1.444E-073.324E-08
c71.291E-091.426E-074.426E-071.309E-08-5.04E-09
c87.522E-111.642E-094.250E-083.532E-10-2.979E-10
c99.165E-131.196E-094.373E-09-6.520E-114.790E-12
c104.275E-145.245E-119.754E-10-5.20E-136.51E-13

6 Conclusions

In this manuscript, we have proposed a modified approach to solve nonlinear Lane-Emden equation using first kind chebyshev operational matrix of differentiation. The benefit of this approach over other existing methods is that only operation matrix of differentiation have need of present the solution more precisely. This differentiation matrix contains mostly number of zeros, which reduces the run time and provides simplicity in computation. This algorithm generates nonlinear system of equations, which can be performed on computer using a program run in MATLAB2016. N must be selected large sufficient to solve the Lane-Emden equation most precisely.

Consequently, the numerical solution concurs with the exact and series solution even with a small number of first kind of chebyshev polynomial used in estimation. Several examples have been worked out to reveal the accuracy and performance of modified algorithm than other recently developed methods.

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Received: 2017-11-26
Revised: 2018-01-16
Accepted: 2018-02-21
Published Online: 2018-05-22
Published in Print: 2019-01-28

© 2019 B. Sharma et al., published by De Gruyter.

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Artikel in diesem Heft

  1. Chebyshev Operational Matrix Method for Lane-Emden Problem
  2. Concentrating solar power tower technology: present status and outlook
  3. Control of separately excited DC motor with series multi-cells chopper using PI - Petri nets controller
  4. Effect of boundary roughness on nonlinear saturation of Rayleigh-Taylor instability in couple-stress fluid
  5. Effect of Heterogeneity on Imbibition Phenomena in Fluid Flow through Porous Media with Different Porous Materials
  6. Electro-osmotic flow of a third-grade fluid past a channel having stretching walls
  7. Heat transfer effect on MHD flow of a micropolar fluid through porous medium with uniform heat source and radiation
  8. Local convergence for an eighth order method for solving equations and systems of equations
  9. Numerical techniques for behavior of incompressible flow in steady two-dimensional motion due to a linearly stretching of porous sheet based on radial basis functions
  10. Influence of Non-linear Boussinesq Approximation on Natural Convective Flow of a Power-Law Fluid along an Inclined Plate under Convective Thermal Boundary Condition
  11. A reliable analytical approach for a fractional model of advection-dispersion equation
  12. Mass transfer around a slender drop in a nonlinear extensional flow
  13. Hydromagnetic Flow of Heat and Mass Transfer in a Nano Williamson Fluid Past a Vertical Plate With Thermal and Momentum Slip Effects: Numerical Study
  14. A Study on Convective-Radial Fins with Temperature-dependent Thermal Conductivity and Internal Heat Generation
  15. An effective technique for the conformable space-time fractional EW and modified EW equations
  16. Fractional variational iteration method for solving time-fractional Newell-Whitehead-Segel equation
  17. New exact and numerical solutions for the effect of suction or injection on flow of nanofluids past a stretching sheet
  18. Numerical investigation of MHD stagnation-point flow and heat transfer of sodium alginate non-Newtonian nanofluid
  19. A New Finance Chaotic System, its Electronic Circuit Realization, Passivity based Synchronization and an Application to Voice Encryption
  20. Analysis of Heat Transfer and Lifting Force in a Ferro-Nanofluid Based Porous Inclined Slider Bearing with Slip Conditions
  21. Application of QLM-Rational Legendre collocation method towards Eyring-Powell fluid model
  22. Hyperbolic rational solutions to a variety of conformable fractional Boussinesq-Like equations
  23. MHD nonaligned stagnation point flow of second grade fluid towards a porous rotating disk
  24. Nonlinear Dynamic Response of an Axially Functionally Graded (AFG) Beam Resting on Nonlinear Elastic Foundation Subjected to Moving Load
  25. Swirling flow of couple stress fluid due to a rotating disk
  26. MHD stagnation point slip flow due to a non-linearly moving surface with effect of non-uniform heat source
  27. Effect of aligned magnetic field on Casson fluid flow over a stretched surface of non-uniform thickness
  28. Nonhomogeneous porosity and thermal diffusivity effects on stability and instability of double-diffusive convection in a porous medium layer: Brinkman Model
  29. Magnetohydrodynamic(MHD) Boundary Layer Flow of Eyring-Powell Nanofluid Past Stretching Cylinder With Cattaneo-Christov Heat Flux Model
  30. On the connection coefficients and recurrence relations arising from expansions in series of modified generalized Laguerre polynomials: Applications on a semi-infinite domain
  31. An adaptive mesh method for time dependent singularly perturbed differential-difference equations
  32. On stretched magnetic flow of Carreau nanofluid with slip effects and nonlinear thermal radiation
  33. Rational exponential solutions of conformable space-time fractional equal-width equations
  34. Simultaneous impacts of Joule heating and variable heat source/sink on MHD 3D flow of Carreau-nanoliquids with temperature dependent viscosity
  35. Effect of magnetic field on imbibition phenomenon in fluid flow through fractured porous media with different porous material
  36. Impact of ohmic heating on MHD mixed convection flow of Casson fluid by considering Cross diffusion effect
  37. Mathematical Modelling Comparison of a Reciprocating, a Szorenyi Rotary, and a Wankel Rotary Engine
  38. Surface roughness effect on thermohydrodynamic analysis of journal bearings lubricated with couple stress fluids
  39. Convective conditions and dissipation on Tangent Hyperbolic fluid over a chemically heating exponentially porous sheet
  40. Unsteady Carreau-Casson fluids over a radiated shrinking sheet in a suspension of dust and graphene nanoparticles with non-Fourier heat flux
  41. An efficient numerical algorithm for solving system of Lane–Emden type equations arising in engineering
  42. New numerical method based on Generalized Bessel function to solve nonlinear Abel fractional differential equation of the first kind
  43. Numerical Study of Viscoelastic Micropolar Heat Transfer from a Vertical Cone for Thermal Polymer Coating
  44. Analysis of Bifurcation and Chaos of the Size-dependent Micro–plate Considering Damage
  45. Non-Similar Comutational Solutions for Double-Diffusive MHD Transport Phenomena for Non-Newtnian Nanofluid From a Horizontal Circular Cylinder
  46. Mathematical model on distributed denial of service attack through Internet of things in a network
  47. Postbuckling behavior of functionally graded CNT-reinforced nanocomposite plate with interphase effect
  48. Study of Weakly nonlinear Mass transport in Newtonian Fluid with Applied Magnetic Field under Concentration/Gravity modulation
  49. MHD slip flow of chemically reacting UCM fluid through a dilating channel with heat source/sink
  50. A Study on Non-Newtonian Transport Phenomena in Mhd Fluid Flow From a Vertical Cone With Navier Slip and Convective Heating
  51. Penetrative convection in a fluid saturated Darcy-Brinkman porous media with LTNE via internal heat source
  52. Traveling wave solutions for (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity
  53. Semitrailer Steering Control for Improved Articulated Vehicle Manoeuvrability and Stability
  54. Thermomechanical nonlinear stability of pressure-loaded CNT-reinforced composite doubly curved panels resting on elastic foundations
  55. Combination synchronization of fractional order n-chaotic systems using active backstepping design
  56. Vision-Based CubeSat Closed-Loop Formation Control in Close Proximities
  57. Effect of endoscope on the peristaltic transport of a couple stress fluid with heat transfer: Application to biomedicine
  58. Unsteady MHD Non-Newtonian Heat Transfer Nanofluids with Entropy Generation Analysis
  59. Mathematical Modelling of Hydromagnetic Casson non-Newtonian Nanofluid Convection Slip Flow from an Isothermal Sphere
  60. Influence of Joule Heating and Non-Linear Radiation on MHD 3D Dissipating Flow of Casson Nanofluid past a Non-Linear Stretching Sheet
  61. Radiative Flow of Third Grade Non-Newtonian Fluid From A Horizontal Circular Cylinder
  62. Application of Bessel functions and Jacobian free Newton method to solve time-fractional Burger equation
  63. A reliable algorithm for time-fractional Navier-Stokes equations via Laplace transform
  64. A multiple-step adaptive pseudospectral method for solving multi-order fractional differential equations
  65. A reliable numerical algorithm for a fractional model of Fitzhugh-Nagumo equation arising in the transmission of nerve impulses
  66. The expa function method and the conformable time-fractional KdV equations
  67. Comment on the paper: “Thermal radiation and chemical reaction effects on boundary layer slip flow and melting heat transfer of nanofluid induced by a nonlinear stretching sheet, M.R. Krishnamurthy, B.J. Gireesha, B.C. Prasannakumara, and Rama Subba Reddy Gorla, Nonlinear Engineering 2016, 5(3), 147-159”
  68. Three-Dimensional Boundary layer Flow and Heat Transfer of a Fluid Particle Suspension over a Stretching Sheet Embedded in a Porous Medium
  69. MHD three dimensional flow of Oldroyd-B nanofluid over a bidirectional stretching sheet: DTM-Padé Solution
  70. MHD Convection Fluid and Heat Transfer in an Inclined Micro-Porous-Channel
https://doi.org/10.1515/nleng-2017-0157
Schlagwörter für diesen Artikel
Lane-Emden equations; chebyshev first kind operational matrix of differentiation; Isothermal gas spheres equation; Nonlinear ODE
Creative Commons
BY 4.0

Artikel in diesem Heft

  1. Chebyshev Operational Matrix Method for Lane-Emden Problem
  2. Concentrating solar power tower technology: present status and outlook
  3. Control of separately excited DC motor with series multi-cells chopper using PI - Petri nets controller
  4. Effect of boundary roughness on nonlinear saturation of Rayleigh-Taylor instability in couple-stress fluid
  5. Effect of Heterogeneity on Imbibition Phenomena in Fluid Flow through Porous Media with Different Porous Materials
  6. Electro-osmotic flow of a third-grade fluid past a channel having stretching walls
  7. Heat transfer effect on MHD flow of a micropolar fluid through porous medium with uniform heat source and radiation
  8. Local convergence for an eighth order method for solving equations and systems of equations
  9. Numerical techniques for behavior of incompressible flow in steady two-dimensional motion due to a linearly stretching of porous sheet based on radial basis functions
  10. Influence of Non-linear Boussinesq Approximation on Natural Convective Flow of a Power-Law Fluid along an Inclined Plate under Convective Thermal Boundary Condition
  11. A reliable analytical approach for a fractional model of advection-dispersion equation
  12. Mass transfer around a slender drop in a nonlinear extensional flow
  13. Hydromagnetic Flow of Heat and Mass Transfer in a Nano Williamson Fluid Past a Vertical Plate With Thermal and Momentum Slip Effects: Numerical Study
  14. A Study on Convective-Radial Fins with Temperature-dependent Thermal Conductivity and Internal Heat Generation
  15. An effective technique for the conformable space-time fractional EW and modified EW equations
  16. Fractional variational iteration method for solving time-fractional Newell-Whitehead-Segel equation
  17. New exact and numerical solutions for the effect of suction or injection on flow of nanofluids past a stretching sheet
  18. Numerical investigation of MHD stagnation-point flow and heat transfer of sodium alginate non-Newtonian nanofluid
  19. A New Finance Chaotic System, its Electronic Circuit Realization, Passivity based Synchronization and an Application to Voice Encryption
  20. Analysis of Heat Transfer and Lifting Force in a Ferro-Nanofluid Based Porous Inclined Slider Bearing with Slip Conditions
  21. Application of QLM-Rational Legendre collocation method towards Eyring-Powell fluid model
  22. Hyperbolic rational solutions to a variety of conformable fractional Boussinesq-Like equations
  23. MHD nonaligned stagnation point flow of second grade fluid towards a porous rotating disk
  24. Nonlinear Dynamic Response of an Axially Functionally Graded (AFG) Beam Resting on Nonlinear Elastic Foundation Subjected to Moving Load
  25. Swirling flow of couple stress fluid due to a rotating disk
  26. MHD stagnation point slip flow due to a non-linearly moving surface with effect of non-uniform heat source
  27. Effect of aligned magnetic field on Casson fluid flow over a stretched surface of non-uniform thickness
  28. Nonhomogeneous porosity and thermal diffusivity effects on stability and instability of double-diffusive convection in a porous medium layer: Brinkman Model
  29. Magnetohydrodynamic(MHD) Boundary Layer Flow of Eyring-Powell Nanofluid Past Stretching Cylinder With Cattaneo-Christov Heat Flux Model
  30. On the connection coefficients and recurrence relations arising from expansions in series of modified generalized Laguerre polynomials: Applications on a semi-infinite domain
  31. An adaptive mesh method for time dependent singularly perturbed differential-difference equations
  32. On stretched magnetic flow of Carreau nanofluid with slip effects and nonlinear thermal radiation
  33. Rational exponential solutions of conformable space-time fractional equal-width equations
  34. Simultaneous impacts of Joule heating and variable heat source/sink on MHD 3D flow of Carreau-nanoliquids with temperature dependent viscosity
  35. Effect of magnetic field on imbibition phenomenon in fluid flow through fractured porous media with different porous material
  36. Impact of ohmic heating on MHD mixed convection flow of Casson fluid by considering Cross diffusion effect
  37. Mathematical Modelling Comparison of a Reciprocating, a Szorenyi Rotary, and a Wankel Rotary Engine
  38. Surface roughness effect on thermohydrodynamic analysis of journal bearings lubricated with couple stress fluids
  39. Convective conditions and dissipation on Tangent Hyperbolic fluid over a chemically heating exponentially porous sheet
  40. Unsteady Carreau-Casson fluids over a radiated shrinking sheet in a suspension of dust and graphene nanoparticles with non-Fourier heat flux
  41. An efficient numerical algorithm for solving system of Lane–Emden type equations arising in engineering
  42. New numerical method based on Generalized Bessel function to solve nonlinear Abel fractional differential equation of the first kind
  43. Numerical Study of Viscoelastic Micropolar Heat Transfer from a Vertical Cone for Thermal Polymer Coating
  44. Analysis of Bifurcation and Chaos of the Size-dependent Micro–plate Considering Damage
  45. Non-Similar Comutational Solutions for Double-Diffusive MHD Transport Phenomena for Non-Newtnian Nanofluid From a Horizontal Circular Cylinder
  46. Mathematical model on distributed denial of service attack through Internet of things in a network
  47. Postbuckling behavior of functionally graded CNT-reinforced nanocomposite plate with interphase effect
  48. Study of Weakly nonlinear Mass transport in Newtonian Fluid with Applied Magnetic Field under Concentration/Gravity modulation
  49. MHD slip flow of chemically reacting UCM fluid through a dilating channel with heat source/sink
  50. A Study on Non-Newtonian Transport Phenomena in Mhd Fluid Flow From a Vertical Cone With Navier Slip and Convective Heating
  51. Penetrative convection in a fluid saturated Darcy-Brinkman porous media with LTNE via internal heat source
  52. Traveling wave solutions for (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity
  53. Semitrailer Steering Control for Improved Articulated Vehicle Manoeuvrability and Stability
  54. Thermomechanical nonlinear stability of pressure-loaded CNT-reinforced composite doubly curved panels resting on elastic foundations
  55. Combination synchronization of fractional order n-chaotic systems using active backstepping design
  56. Vision-Based CubeSat Closed-Loop Formation Control in Close Proximities
  57. Effect of endoscope on the peristaltic transport of a couple stress fluid with heat transfer: Application to biomedicine
  58. Unsteady MHD Non-Newtonian Heat Transfer Nanofluids with Entropy Generation Analysis
  59. Mathematical Modelling of Hydromagnetic Casson non-Newtonian Nanofluid Convection Slip Flow from an Isothermal Sphere
  60. Influence of Joule Heating and Non-Linear Radiation on MHD 3D Dissipating Flow of Casson Nanofluid past a Non-Linear Stretching Sheet
  61. Radiative Flow of Third Grade Non-Newtonian Fluid From A Horizontal Circular Cylinder
  62. Application of Bessel functions and Jacobian free Newton method to solve time-fractional Burger equation
  63. A reliable algorithm for time-fractional Navier-Stokes equations via Laplace transform
  64. A multiple-step adaptive pseudospectral method for solving multi-order fractional differential equations
  65. A reliable numerical algorithm for a fractional model of Fitzhugh-Nagumo equation arising in the transmission of nerve impulses
  66. The expa function method and the conformable time-fractional KdV equations
  67. Comment on the paper: “Thermal radiation and chemical reaction effects on boundary layer slip flow and melting heat transfer of nanofluid induced by a nonlinear stretching sheet, M.R. Krishnamurthy, B.J. Gireesha, B.C. Prasannakumara, and Rama Subba Reddy Gorla, Nonlinear Engineering 2016, 5(3), 147-159”
  68. Three-Dimensional Boundary layer Flow and Heat Transfer of a Fluid Particle Suspension over a Stretching Sheet Embedded in a Porous Medium
  69. MHD three dimensional flow of Oldroyd-B nanofluid over a bidirectional stretching sheet: DTM-Padé Solution
  70. MHD Convection Fluid and Heat Transfer in an Inclined Micro-Porous-Channel
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