Abstract
To improve the corrosion and mechanical properties of the AM50 magnesium alloy, different amounts of the rare earth element gadolinium were used. The microstructure, corrosion and mechanical properties were evaluated by X-ray diffraction, scanning electron microscopy, energy dispersive spectroscopy, and electrochemical and mechanical stretch methods. The results indicate that, with Gd addition, the amount of the Al2Gd3 phase increased while the β-Mg17Al12 phase amount decreased. Due to the Gd addition, the grain of the AM50 magnesium alloy was significantly refined, which improved its tensile strength. Further, the decrease in the amount of the β phase improved the corrosion resistance of the alloy. The fracture mechanism of the Gd-modified AM50 magnesium alloy was a quasi-cleavage fracture. Finally, the optimum corrosion residual strength of the AM50 magnesium alloy occurred with 1 wt.%of added Gd.
1 Introduction
Magnesium alloys are the lightest structure metal, and have therefore been called “the 21st Century Green Structure Metal” [1]. Magnesium alloys are promising candidates to replace steel and aluminum alloys in many structural and mechanical applications owing to their attractive properties of excellent castability, good machinability, superior damping capacity, outstanding stiffness-to-weight ratio, and ease of recyclability [234]. The AM50 alloy is one of the most successfully-used magnesium alloys in the automotive industry; however, its application is still limited by its strength and most especially its poor corrosion resistance [5, 6].
Typically, corrosion is inevitable with usage, and the loss of the mechanical properties that accompanies the corrosion process is thus also inevitable. Here, the tensile strength of the materials tested after a corrosion test was defined as the corrosion residual strength (CRS), and the study of the CRS of materials is of great significance to predict their service life [7]. Therefore, building an evaluation system for CRS will promote the application of magnesium alloys.
Rare earth elements, sometimes called “metal vitamins” because of their unique attribute of improving the performance of metals, have been used to modify magnesium alloys for several years. For instance, Wang [8], Li [9] and Song [10] have studied the effects of the rare earth elements Y, Ce and Nd upon the corrosion resistance of magnesium alloys. Gadolinium is a prime candidate as a modifying rare earth element for magnesium alloy. The extranuclear electron arrangement of Gd is 4f75d16s2, where the 4f electron orbit is a half-filled state that creates a stable electron structure. The atomic radius of Gd is large, and the two outer-layer s electrons and the single secondouter-layer 5d electron are easy to lose. Further, Gd can become a positive trivalent ion with high chemical activity, which produces a strong binding force with atoms such as O and S. To date, the corrosion and mechanical properties of Gd-modified AM50 magnesium alloys have been investigated infrequently. This paper, therefore, investigates modifying the corrosion and mechanical properties of a cast AM50 magnesium alloy by adding the rare earth element Gd.
2 Experimental
2.1 Preparation of sample
The experimental alloys used in this study were a commercial AM50 Mg alloy, a Mg-Gd master alloy (Gd = 20 wt.%) and industry-pure Al. The casting equipment were a welltype electric resistance furnace and a graphite crucible. First, the AM50 Mg alloy was re-melted in a highly-pure argon environment. When the temperature of the furnace was ~720°C, the Mg-Gd master alloy and pure Al were placed into the melting AM50 Mg alloy liquid. The melting alloy was stirred and standed for 20 min. Then the melting alloy was cast into the mold at the temperature of 710°C . where the size of the cast ingot was 200×130×12 mm3. The samples were denoted as AM50GdX (where X represents the content of Gd), and their compositions are listed in Table 1.
2.2 Tensile test
The specimens used for tensile testing were cut to 120×12×12 mm3 and lathed into their final shape, with the dimensions shown in Fig. 1. The samples were first polished, and then washed and dried. Tensile tests equipment was an AG-10TA materials tensile test machine. The stretching rate was 1 mm min-1. The lowest strength obtained from three identical samples measured under the same corrosion time was recorded as the safety corrosion residual strength if the samples were without any pores or slag.

Diagrammatic sketch of the tensile test sample with its dimension.
Compositions of the Gd-modified AM50 Mg alloys
Sample named | Gd (wt.%) | AM50 (wt.%) Bal. |
---|---|---|
AM50Gd1 | 0.5 | Bal. |
AM50Gd2 | 1.0 | Bal. |
AM50Gd3 | 1.5 | Bal. |
AM50Gd4 | 2.0 | Bal. |
2.3 Corrosion process
The corrosion immersion tests were carried out in a 3.5 wt.% NaCl solution made with analytical reagent chemicals and distilled water. Before testing, an Mg(OH)2 powder was placed in the test solution to obtain a saturated Mg(OH)2 NaCl solution, which could produce a consistent rate of corrosion [7]. The immersion test was performed at 25°C in a constant-temperature chamber for a corrosion time of 24, 72, 168, 264, 336 and 432 h, separately [7]. At each time point, three tensile test samples were removed from the corrosion solution and washed with a chromic-acid silver-nitrate aqueous solution and with absolute ethyl alcohol, and were subsequently dried with cold flowing air. Finally, the tensile test samples were stored in numbered sample bags.
2.4 Metallographic sample process
The samples used for microstructural observation were corroded using 3.5 wt.% nitric acid ethyl alcohol for 30 s. The microstructure, the main view and vertical sections of the corroded samples were recorded and analyzed by a scanning electron microscope (SEM; JSM-5600, JEOL Inc.). A detailed analysis of the phases present within each sample was carried out via X-ray diffractometry (XRD; LabX XRD-6000, Shimadzu).
2.5 Electrochemical polarization test
Electrochemical polarization tests were carried out in a LK98B11 chemical workstation (Lanlike), with an exposed area of 10×10 mm2 and where the test solution was a 3.5 wt.% NaCl aqueous solution. A standard three-electrode configuration was used, wherein saturated calomel was used as a reference electrode, platinum as a standard electrode and the sample as the working electrode. Specimens were immersed in the test solution and a polarization scan was carried out at a rate of 10 mV s-1.
3 Results and discussion
3.1 Microstructure
The XRD spectra of the Gd-modified AM50 magnesium alloy samples are shown in Fig. 2, where the phases existing in the alloy include α-Mg, β-Mg17Al12, Al0.4GdMn1.6 and Al2Gd3. From the Mg-Al phase diagram, the Mg and Al (4.427 wt.%) in the AM50 magnesium alloy during the solidification process are present in the α-Mg and β-Mg17Al12 phases. After Gd was added, the Al, Gd and Mn form Al2Gd3 and Al0.4GdMn1.6, where the diffraction peak intensity of the Al2Gd3 and Al0.4GdMn1.6 phases increases with the addition of Gd while that of the β-Mg17Al12 phase decreases.

X-ray diffraction spectra from the Gd-modified AM50 magnesium alloys. The samples are AM50 Mg alloy modified with Gd in the amount of 0.5 wt.% (AM50Gd1), 1.0 wt.% (AM50Gd2), 1.5 wt.% (AM50Gd3) and 2.0 wt.% (AM50Gd4).
The microstructure morphology of the as-cast AM50 magnesium alloy and the AM50GdX alloys (where X represents the content of Gd) are illustrated using SEM images in Fig. 3. The original microstructure of the as-cast AM50 magnesium alloy comprises primarily the α-Mg phase (medium intensity areas in the SEM image) and the highintensity island-shaped β-Mg17Al12 phase (Fig. 3(a)). After Gd is added, the brighter-intensity small-grain Al2Gd3 and Al0.4GdMn1.6 exist at the grain boundaries (Figs.3(b)–3(e)). With increasing amounts of Gd, the amount of brighter-intensity small-grain Al2Gd3 and Al0.4GdMn1.6 phases increases, while the amount and volume of the β-Mg17Al12 phase decreases. From Fig. 3(b) it can be seen that, when the Gd content is 0.5 wt.%, the amount of β phase noticeably decreases and the visible Al2Gd3 and Al0.4GdMn1.6 phases are few. When 1 wt.% Gd is added, the amount of the phases containing Gd increases and the amount of the β phase noticeably decreases (Fig. 3(c)). When 1.5 wt.% Gd is added (Fig. 3(d)), the amount and volume of high-intensity Al2Gd3 and Al0.4GdMn1.6 phases increases further. Finally, with 2.0 wt.% of added Gd (Fig. 3(e)), the β-phase areas become as small as the areas containing the Al2Gd3 and Al0.4GdMn1.6 phases.

Scanning electron microscope images of the morphologies of the magnesium alloys. The sample alloys comprise the (a) as-cast AM50, (b) AM50Gd1 containing 0.5 wt.% Gd, (c) AM50Gd2 containing 1.0 wt.% Gd, (d) AM50Gd3 containing 1.5 wt.% Gd, and (e) AM50Gd4 containing 2.0 wt.% Gd.
The SEM morphology of AM50Gd2 and the energy dispersive spectroscopy (EDS) results from two points (labeled A and B) are shown in Fig. 4c. Under 5000× magnification, many smaller areas of the β phase are observed, and the Al0.4GdMn1.6 and Al2Gd3 phases are seen to accompany one another. During the solidification process, Gd exhibits a preference for combining with Al over Mg [11]. Owing to its higher melting point than that of the α-Mg and β-Mg17Al12 phases, the high-strength and thermally stable Al2Gd3 phase separates out first at the front of the solidification interface. This inhibits the separation of the β-Mg17Al12 phase and refines the microstructure of the alloy. The rare earth element Gd, therefore, improves the nucleation rate and leads to grain refinement and grain boundary strengthening of the AM50 magnesium alloy.

Scanning electron microscope images of the morphology and energy dispersive X-ray spectra of the AM50Gd2 sample containing 1.0 wt.% added Gd. The image (left) shows the points from where the spectra for point A (upper right) and point B (lower right) are obtained.
3.2 Mechanical character
Figure 5 shows the engineering stress-strain curves of the AM50GdX magnesium alloys. The tensile strength of the AM50 magnesium alloy after Gd modification is found to increase with Gd addition, with the values 210 MPa (Gd = 0.5 wt.%), 214 MPa (Gd = 1.0 wt.%), 215 MPa (Gd = 1.5 wt.%) and 220 MPa (Gd = 2.0 wt.%). It can be observed that the curves from the varying alloys are similar in shape and consist of the two ranges of linear elastic and plastic zones, where the plastic zone is markedly longer than the linear zone. With the increase of Gd content, the slopes of the curves initially increase and then decrease. The incorporation of rare earth element Gd causes grain refinement strengthening and grain boundary strengthening, which improves the tensile properties of the AM50 magnesium alloy. However, excessive quantities of Gd lead to a reduction in the slope of the engineering stress-strain curve, which means that the elongation of the alloy is improved via the plastic deformation properties and that the toughness is reduced at the same time. Further, the shape of the stress-strain curves reveals that the fracture mode of the AM50GdX magnesium alloy is quasi-cleavage fracture.

The engineering stress-strain curves of AM50GdX magnesium alloys. The samples are AM50 Mg alloy modified with Gd in the amount of 0.5 wt.% (AM50Gd1), 1.0 wt.%
3.3 Corrosion character
Figure 6 shows Tafel curves of AM50GdX magnesium alloy, where the corrosion potentials of the AM50Gd1–AM50Gd4 magnesium alloys are -1.65, -1.75, -1.72 and -1.71 V, respectively. In this range, with an increasing amount of Gd, the corrosion potential initially decreases and then increases. Simultaneously, the corrosion current density of the cathodic branch initially decreases and then increases. The turning point for both of these parameters is 1.0 wt.% Gd (AM50Gd2). Owing to the significant difference between the Gd and Mg atoms, it is difficult to solidsolute the Gd atom into the AM50 magnesium alloy, but instead the Gd tends to form rare earth compounds. When the Gd amount is less than 1.0 wt.%, the effect of Gd addition is grain refinement, which reduces the micro-galvanic effect [7, 12]. However, with more Gd the amount and volume of the Al2Gd3 phase significantly increases, which leads to an aggravation of the micro-galvanic effect and a hydrogen evolution reaction. This explains why corrosion resistance worsens after adding more than 1 wt.% Gd.

Tafel curves of the AM50GdX magnesium alloys. The samples are AM50 Mg alloy modified with Gd at 0.5 wt.% (AM50Gd1), 1.0 wt.%
The surface corrosion morphologies of the AM50GdX samples after a 24 h immersion in the corrosion solution are shown in Fig. 7. Corrosion pitting appears on all of the sample surfaces after 24 h immersion in NaCl aqueous solution, but the amount of corrosion pitting still follows the trend of initially decreasing and then increasing as the Gd amount increases. Interestingly, the turning point for the maximum pitting also appears at AM50Gd2 (Gd = 1 wt.%). During the corrosion process in aqueous solution, the oxide or hydroxide film on the surface of the magnesium alloy protects the magnesium alloy matrix, and so the quality of the oxide or hydroxide film determines the corrosion pitting nucleation rate. The rare earth element Gd contributes to the formation of a high-quality oxide or hydroxide film, but when the Gd addition is over 1.0 wt.% the quality of the oxide or hydroxide film worsens. The addition of excessive amounts of Gd leads to an increase of the Al-Gd phase, which destroys the oxide or hydroxide film, and the corrosion pitting and corrosion current begins to increase. Because the atoms of the rare earth element tend to be attracted to each other [13], they form atomic clusters that lead to the Al-Gd phase segregation. Figure 7 (bottom right labeled AM50Gd4) gives evidence of the corrosion pitting concentrated on a localized region of this sample containing Gd = 2.0 wt.%.

The surface corrosion morphology of the AM50GdX samples after 24 h immersion in the corrosion solution. The samples shown are AM50 Mg alloy modified with Gd at 0.5 wt.% (AM50Gd1, upper left), 1.0 wt.% (AM50Gd2, upper right), 1.5 wt.% (AM50Gd3, lower left) and 2.0 wt.% (AM50Gd4, lower right).
3.4 Corrosion residual strength(CRS)
Figure 8 shows the corrosion residual strength (CRS) curves of the AM50GdX magnesium alloys where it can be seen that, for a corrosion time greater than 168 h, the CRS value of the AM50GdX samples decreases linearly. Further, at each time point above 168 h the CRS value of the AM50Gd2 sample is greater than that of the others, signifying that 1 wt.% Gd addition is beneficial for improving the CRS of AM50 magnesium. In the CRS curve of AM50Gd1 (Gd = 0.5 wt.%), the reduction rate of the CRS value showed that there are two interregional: the beginning of the corrosion process for this sample the corrosion rate is higher, but after about 150 h into the corrosion process the corrosion rate reduces. This is because, after 150 h of immersion in the NaCl aqueous solution, the oxide or hydroxide film formed on the surface of the AM50Gd1 sample is in a stable state, and the type of corrosion is therefore a uniform corrosion [14]. The CRS curves of the AM50Gd2, AM50Gd3 and AM50Gd4 samples exhibit a different trend, however, wherein the CRS value reduces in a linear fashion throughout the corrosion process. Therefore, the depth of the corrosion pitting on the AM50Gd2, AM50Gd3 and AM50Gd4 samples increases until the steady stage.

The corrosion residual strength (ςCRS) vs. corrosion time for the AM50GdX magnesium alloys. The samples are AM50 Mg alloy modified with Gd at 0.5 wt.% (AM50Gd1), 1.0 wt.% (AM50Gd2), 1.5 wt.% (AM50Gd3) and 2.0 wt.% (AM50Gd4).
In previous study [7], it is found that the existence and development of corrosion pits are the direct reason for the dropping of the tensile strength. First, corrosion pits present leads to a decrease of the CRS value directly because of the decrease of the cross-sectional area. Second, the stress concentration at the large change of curvature in the corrosion pits and the emergence of micro-cracks at the bottom of the corrosion pits causes fracture of the tensile samples. Through statistics and calculations, the stress concentration is more important than the depth of the corrosion pits. Therefore, reducing the corrosion pits is beneficial for improving the CRS of these AM50GdX samples.
With the tensile test samples immersed in the corrosion solution, the external and internal corrosion factors of the magnesium alloy interact at the surface of the samples [15], which leads to the nucleation and growth of the corrosion pitting. The presence of Gd contributes in two ways to affect the CRS value of the AM50 magnesium alloy, via the strengthening of its tensile properties and the improvement of its corrosion resistance. First, the methods whereby the AM50 magnesium alloy is strengthened by Gd addition are owing to grain refinement, grain boundary and second-phase strengthening. Second, Gd improves the corrosion resistance. In the corrosion process, corrosion pitting nucleates at the α-Mg phase because the corrosion potential of the β phase and the Al-Gd phase are higher than that of α-Mg phase [7]. Therefore, after Gd addition the nucleation of corrosion pitting becomes more difficult.
4 Conclusions
The corrosion and mechanical performance of a Gdmodified AM50 magnesium alloy have been studied. It was found that the phases within the Gd-modified AM50 magnesium alloy include the α-Mg, β-Mg17Al12, Al2Gd3 and Al0.4GdMn1.6 phases. The rare earth element Gd contributes toward grain refinement, grain boundary strengthening and second-phase strengthening to improve the tensile strength of the alloy. Further, Gd also improves the corrosion resistance of the AM50 magnesium alloy by reducing the β-Mg17Al12 phase. However, with increasing amounts of added Gd, the corrosion resistance of AM50 initially improves and then worsens, where the turning point for this corrosion resistance is at 1.0 wt.% Gd addition. It is therefore concluded that the corrosion residual strength of the 1.0 wt.% Gd-modified AM50 magnesium alloy (AM50Gd2) is higher than that of the other AM50GdX alloys tested.
Acknowledgement
This study was supported by the Science and Technology Development Projects of Changchun City (No.11KZ31), the National Natural Science Foundation of China(No.51201068), Science and Technology Development Projects of Jilin Province (No. 201201032) and the Fundamental Research Funds for the Central Universities (Jilin University, Grant No. 2013ZY04)
References
[1] Yang M., Liu Y.H., Jin H.Z., Song Y.L., Influence of solid-solution treatment on tensile properties of cast AM50 magnesium alloy after corrosion tests, J. Tran. Mater. heat treat., 2012, 10.13289/j.issn.1009-6264.2012.07.020Search in Google Scholar
[2] Zeng R.C., Chen J., Dietzle W., Zettler R. Santos J.F., Nascimento M.L., et al., Corrosion of friction stir welded magnesium alloy AM50, J.Cor. Sci., 2009, http://www.sciencedirect.com/scie nce/article/pii/S0010938X09001875/scie nce/article/pii/S0010938X09001875Search in Google Scholar
[3] Shi Z.M., Song G.L., Atrens A., Influence of anodising current on the corrosion resistance of anodised AZ91D magnesium alloy, J. Cor. sci., 2006, http://www.sciencedirect.com/science/article/pii/S0010938X0500243X10.1016/j.corsci.2005.08.004Search in Google Scholar
[4] Wu G.H., Fan Y., Gao H.T., Zhai C.Q., Zhu Y.P., The effect of Ca and rare earth elements on the microstructure, mechanical properties and corrosion behavior of AZ91D, J. Mater. Sci. Eng.A,2005, http://www.sciencedirect.com/science/article/pii/S092150930500934210.1016/j.msea.2005.08.011Search in Google Scholar
[5] Zhou W., Shen T., Aung N.N., Effect of heat treatment on corrosion behaviour of magnesium alloy AZ91D in simulated body fluid, J. Cor. Sci., 2010, http://www.sciencedirect.com/science/article/pii/S0010938X0900584810.1016/j.corsci.2009.11.030Search in Google Scholar
[6] Pardo A., Merino M.C., Coy A.E., Arrabal R., Viejo F., Matykina E., Corrosion behaviour of magnesium/aluminium alloys in 3.5 wt.% NaCl, J. Cor. Sci., 2008, http://www.sciencedirect.com/science/article/pii/S0010938X0700312510.1016/j.corsci.2007.11.005Search in Google Scholar
[7] Yang M., Liu Y.H., Liu J.A., Song Y.L., Effect of T6 heat treatment on corrosion resistance and mechanical properties of AM50 magnesium alloy, Mater. Res. Innovat., 2015, http://www.tandfonline.com/doi/abs/10.1179/1432891715Z.000000000216010.1179/1432891715Z.0000000002160Search in Google Scholar
[8] Wang Q., Liu Y.H., Zhu X.Y., Li S., Yu S.R., Zhang L.N., et al., Study on the effect of corrosion on the tensile properties of the 1.0 wt.% Yttrium modified AZ91 magnesium alloy, J. Mat. Sci. Eng. A, 2009, http://www.sciencedirect.com/science/article/pii/S092150930900402X10.1016/j.msea.2009.03.053Search in Google Scholar
[9] Li C.F., Liu Y.H., Wang Q., Zhang L.N., Zhang D.W., Study on the corrosion residual strength of the 1.0 wt.% Ce modified AZ91 magnesium alloy, J. Mater. Char., 2009, http://www.sciencedirect.com/science/article/pii/S104458030900292710.1016/j.matchar.2009.09.014Search in Google Scholar
[10] Song, Y.L., Liu, Y.H., Yu, S.R., Zhu X.Y., Wang S.H., Effect of neodymium on microstructure and corrosion resistance of AZ91 magnesium alloy, J. Mater. Sci., 2007, 10.1007/s10853-006-0661-zSearch in Google Scholar
[11] Wang L., Zhang B.P., Shinohara T., Corrosion behavior of AZ91 magnesium alloy in dilute NaCl solutions, Mater. Des., 2010, http://www.sciencedirect.com/science/article/pii/S026130690900390210.1016/j.matdes.2009.07.049Search in Google Scholar
[12] Song G.L., Bowles A.L., StJohn D.H., Corrosion resistance of aged die cast magnesium alloy AZ91D, J. Mat. Sci. Eng. A, 2004, http://www.sciencedirect.com/science/article/pii/S092150930300817710.1016/j.msea.2003.08.060Search in Google Scholar
[13] Liu G.L., Electronic theoretical study on the influence of rare earth on the stress corrosion in magnesium alloy, J. Acta Phy. sin., 2006, http://www.cnki.net/KCMS/detail/det ail.aspx?QueryID=15&CurRec=7&recid=&filename=WLXB200612063&dbname=CJFD2006&dbcode=CJFQ&pr=&urlid=&yx=&uid=WEEvREcwSlJHSldRa1Fhb09jeVVYampBSGJZQjY5VVBiTkdR RXRFL1M2QT0=$9A4hF_YAuvQ5obgVAqNKPCYcEjKensW4ggI8Fm4gTkoUKaID8j8gFw!!&v=MDc2MTZWNy9LTWlIVGJMRzRIdGZOclk5RFo0UjhlWDFMdXhZUzdEaDFUM3FUcldNMUZyQ1VSTHllWitkdEZ5M20=Search in Google Scholar
[14] Song D., Ma A.B., Jiang J.H., Lin P.H., Yang D.H., Fan J.F., Corrosion behavior of equal-channel-angular-pressed pure magnesium in NaCl aqueous solution, J. Cor. Sci. 2010, http://www.sciencedirect.com/science/article/pii/S0010938X0900487910.1016/j.corsci.2009.10.004Search in Google Scholar
[15] Liu W.J., Cao F.H., Chang L.R., Zhang Z., Zhang J.Q., Effect of rare earth element Ce and La on corrosion behavior of AM60 magnesium alloy, J. Cor. Sci., 2009, http://www.sciencedirect.com/science/article/pii/S0010938X0900110310.1016/j.corsci.2009.03.018Search in Google Scholar
© 2016 M. Yang et al.
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.
Articles in the same Issue
- Regular articles
- Speeding of α Decay in Strong Laser Fields
- Regular articles
- Multi-soliton rational solutions for some nonlinear evolution equations
- Regular articles
- Thin film flow of an Oldroyd 6-constant fluid over a moving belt: an analytic approximate solution
- Regular articles
- Bilinearization and new multi-soliton solutions of mKdV hierarchy with time-dependent coefficients
- Regular articles
- Duality relation among the Hamiltonian structures of a parametric coupled Korteweg-de Vries system
- Regular articles
- Modeling the potential energy field caused by mass density distribution with Eton approach
- Regular articles
- Climate Solutions based on advanced scientific discoveries of Allatra physics
- Regular articles
- Investigation of TLD-700 energy response to low energy x-ray encountered in diagnostic radiology
- Regular articles
- Synthesis of Pt nanowires with the participation of physical vapour deposition
- Regular articles
- Quantum discord and entanglement in grover search algorithm
- Regular articles
- On order statistics from nonidentical discrete random variables
- Regular articles
- Charmed hadron photoproduction at COMPASS
- Regular articles
- Perturbation solutions for a micropolar fluid flow in a semi-infinite expanding or contracting pipe with large injection or suction through porous wall
- Regular articles
- Flap motion of helicopter rotors with novel, dynamic stall model
- Regular articles
- Impact of severe cracked germanium (111) substrate on aluminum indium gallium phosphate light-emitting-diode’s electro-optical performance
- Regular articles
- Slow-fast effect and generation mechanism of brusselator based on coordinate transformation
- Regular articles
- Space-time spectral collocation algorithm for solving time-fractional Tricomi-type equations
- Regular articles
- Recent Progress in Search for Dark Sector Signatures
- Regular articles
- Recent progress in organic spintronics
- Regular articles
- On the Construction of a Surface Family with Common Geodesic in Galilean Space G3
- Regular articles
- Self-healing phenomena of graphene: potential and applications
- Regular articles
- Viscous flow and heat transfer over an unsteady stretching surface
- Regular articles
- Spacetime Exterior to a Star: Against Asymptotic Flatness
- Regular articles
- Continuum dynamics and the electromagnetic field in the scalar ether theory of gravitation
- Regular articles
- Corrosion and mechanical properties of AM50 magnesium alloy after modified by different amounts of rare earth element Gadolinium
- Regular articles
- Genocchi Wavelet-like Operational Matrix and its Application for Solving Non-linear Fractional Differential Equations
- Regular articles
- Energy and Wave function Analysis on Harmonic Oscillator Under Simultaneous Non-Hermitian Transformations of Co-ordinate and Momentum: Iso-spectral case
- Regular articles
- Unification of all hyperbolic tangent function methods
- Regular articles
- Analytical solution for the correlator with Gribov propagators
- Regular articles
- A New Algorithm for the Approximation of the Schrödinger Equation
- Regular articles
- Analytical solutions for the fractional diffusion-advection equation describing super-diffusion
- Regular articles
- On the fractional differential equations with not instantaneous impulses
- Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
- Exact solutions of the Biswas-Milovic equation, the ZK(m,n,k) equation and the K(m,n) equation using the generalized Kudryashov method
- Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
- Numerical solution of two dimensional time fractional-order biological population model
- Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
- Rotational surfaces in isotropic spaces satisfying weingarten conditions
- Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
- Anti-synchronization of fractional order chaotic and hyperchaotic systems with fully unknown parameters using modified adaptive control
- Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
- Approximate solutions to the nonlinear Klein-Gordon equation in de Sitter spacetime
- Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
- Stability and Analytic Solutions of an Optimal Control Problem on the Schrödinger Lie Group
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- Logical entropy of quantum dynamical systems
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- An efficient algorithm for solving fractional differential equations with boundary conditions
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- A numerical method for solving systems of higher order linear functional differential equations
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- Nonlinear self adjointness, conservation laws and exact solutions of ill-posed Boussinesq equation
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- On combined optical solitons of the one-dimensional Schrödinger’s equation with time dependent coefficients
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- On soliton solutions of the Wu-Zhang system
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- Comparison between the (G’/G) - expansion method and the modified extended tanh method
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- On the union of graded prime ideals
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- Oscillation criteria for nonlinear fractional differential equation with damping term
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- A new method for computing the reliability of consecutive k-out-of-n:F systems
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- A time-delay equation: well-posedness to optimal control
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- Numerical solutions of multi-order fractional differential equations by Boubaker polynomials
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- Laplace homotopy perturbation method for Burgers equation with space- and time-fractional order
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- The calculation of the optical gap energy of ZnXO (X = Bi, Sn and Fe)
- Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
- Analysis of time-fractional hunter-saxton equation: a model of neumatic liquid crystal
- Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
- A certain sequence of functions involving the Aleph function
- Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
- On negacyclic codes over the ring ℤp + uℤp + . . . + uk + 1 ℤp
- Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
- Solitary and compacton solutions of fractional KdV-like equations
- Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
- Regarding on the exact solutions for the nonlinear fractional differential equations
- Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
- Non-local Integrals and Derivatives on Fractal Sets with Applications
- Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
- On the solutions of electrohydrodynamic flow with fractional differential equations by reproducing kernel method
- Special issue on Information Technology and Computational Physics
- On uninorms and nullnorms on direct product of bounded lattices
- Special issue on Information Technology and Computational Physics
- Phase-space description of the coherent state dynamics in a small one-dimensional system
- Special issue on Information Technology and Computational Physics
- Automated Program Design – an Example Solving a Weather Forecasting Problem
- Special issue on Information Technology and Computational Physics
- Stress - Strain Response of the Human Spine Intervertebral Disc As an Anisotropic Body. Mathematical Modeling and Computation
- Special issue on Information Technology and Computational Physics
- Numerical solution to the Complex 2D Helmholtz Equation based on Finite Volume Method with Impedance Boundary Conditions
- Special issue on Information Technology and Computational Physics
- Application of Genetic Algorithm and Particle Swarm Optimization techniques for improved image steganography systems
- Special issue on Information Technology and Computational Physics
- Intelligent Chatter Bot for Regulation Search
- Special issue on Information Technology and Computational Physics
- Modeling and optimization of Quality of Service routing in Mobile Ad hoc Networks
- Special issue on Information Technology and Computational Physics
- Resource management for server virtualization under the limitations of recovery time objective
- Special issue on Information Technology and Computational Physics
- MODY – calculation of ordered structures by symmetry-adapted functions
- Special issue on Information Technology and Computational Physics
- Survey of Object-Based Data Reduction Techniques in Observational Astronomy
- Special issue on Information Technology and Computational Physics
- Optimization of the prediction of second refined wavelet coefficients in electron structure calculations
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Droplet spreading and permeating on the hybrid-wettability porous substrates: a lattice Boltzmann method study
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- POD-Galerkin Model for Incompressible Single-Phase Flow in Porous Media
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Effect of the Pore Size Distribution on the Displacement Efficiency of Multiphase Flow in Porous Media
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Numerical heat transfer analysis of transcritical hydrocarbon fuel flow in a tube partially filled with porous media
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Experimental Investigation on Oil Enhancement Mechanism of Hot Water Injection in tight reservoirs
- Special Issue on Research Frontier on Molecular Reaction Dynamics
- Role of intramolecular hydrogen bonding in the excited-state intramolecular double proton transfer (ESIDPT) of calix[4]arene: A TDDFT study
- Special Issue on Research Frontier on Molecular Reaction Dynamics
- Hydrogen-bonding study of photoexcited 4-nitro-1,8-naphthalimide in hydrogen-donating solvents
- Special Issue on Research Frontier on Molecular Reaction Dynamics
- The Interaction between Graphene and Oxygen Atom
- Special Issue on Research Frontier on Molecular Reaction Dynamics
- Kinetics of the austenitization in the Fe-Mo-C ternary alloys during continuous heating
- Special Issue: Functional Advanced and Nanomaterials
- Colloidal synthesis of Culn0.75Ga0.25Se2 nanoparticles and their photovoltaic performance
- Special Issue: Functional Advanced and Nanomaterials
- Positioning and aligning CNTs by external magnetic field to assist localised epoxy cure
- Special Issue: Functional Advanced and Nanomaterials
- Quasi-planar elemental clusters in pair interactions approximation
- Special Issue: Functional Advanced and Nanomaterials
- Variable Viscosity Effects on Time Dependent Magnetic Nanofluid Flow past a Stretchable Rotating Plate
Articles in the same Issue
- Regular articles
- Speeding of α Decay in Strong Laser Fields
- Regular articles
- Multi-soliton rational solutions for some nonlinear evolution equations
- Regular articles
- Thin film flow of an Oldroyd 6-constant fluid over a moving belt: an analytic approximate solution
- Regular articles
- Bilinearization and new multi-soliton solutions of mKdV hierarchy with time-dependent coefficients
- Regular articles
- Duality relation among the Hamiltonian structures of a parametric coupled Korteweg-de Vries system
- Regular articles
- Modeling the potential energy field caused by mass density distribution with Eton approach
- Regular articles
- Climate Solutions based on advanced scientific discoveries of Allatra physics
- Regular articles
- Investigation of TLD-700 energy response to low energy x-ray encountered in diagnostic radiology
- Regular articles
- Synthesis of Pt nanowires with the participation of physical vapour deposition
- Regular articles
- Quantum discord and entanglement in grover search algorithm
- Regular articles
- On order statistics from nonidentical discrete random variables
- Regular articles
- Charmed hadron photoproduction at COMPASS
- Regular articles
- Perturbation solutions for a micropolar fluid flow in a semi-infinite expanding or contracting pipe with large injection or suction through porous wall
- Regular articles
- Flap motion of helicopter rotors with novel, dynamic stall model
- Regular articles
- Impact of severe cracked germanium (111) substrate on aluminum indium gallium phosphate light-emitting-diode’s electro-optical performance
- Regular articles
- Slow-fast effect and generation mechanism of brusselator based on coordinate transformation
- Regular articles
- Space-time spectral collocation algorithm for solving time-fractional Tricomi-type equations
- Regular articles
- Recent Progress in Search for Dark Sector Signatures
- Regular articles
- Recent progress in organic spintronics
- Regular articles
- On the Construction of a Surface Family with Common Geodesic in Galilean Space G3
- Regular articles
- Self-healing phenomena of graphene: potential and applications
- Regular articles
- Viscous flow and heat transfer over an unsteady stretching surface
- Regular articles
- Spacetime Exterior to a Star: Against Asymptotic Flatness
- Regular articles
- Continuum dynamics and the electromagnetic field in the scalar ether theory of gravitation
- Regular articles
- Corrosion and mechanical properties of AM50 magnesium alloy after modified by different amounts of rare earth element Gadolinium
- Regular articles
- Genocchi Wavelet-like Operational Matrix and its Application for Solving Non-linear Fractional Differential Equations
- Regular articles
- Energy and Wave function Analysis on Harmonic Oscillator Under Simultaneous Non-Hermitian Transformations of Co-ordinate and Momentum: Iso-spectral case
- Regular articles
- Unification of all hyperbolic tangent function methods
- Regular articles
- Analytical solution for the correlator with Gribov propagators
- Regular articles
- A New Algorithm for the Approximation of the Schrödinger Equation
- Regular articles
- Analytical solutions for the fractional diffusion-advection equation describing super-diffusion
- Regular articles
- On the fractional differential equations with not instantaneous impulses
- Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
- Exact solutions of the Biswas-Milovic equation, the ZK(m,n,k) equation and the K(m,n) equation using the generalized Kudryashov method
- Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
- Numerical solution of two dimensional time fractional-order biological population model
- Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
- Rotational surfaces in isotropic spaces satisfying weingarten conditions
- Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
- Anti-synchronization of fractional order chaotic and hyperchaotic systems with fully unknown parameters using modified adaptive control
- Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
- Approximate solutions to the nonlinear Klein-Gordon equation in de Sitter spacetime
- Topical Issue: Uncertain Differential Equations: Theory, Methods and Applications
- Stability and Analytic Solutions of an Optimal Control Problem on the Schrödinger Lie Group
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- Logical entropy of quantum dynamical systems
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- An efficient algorithm for solving fractional differential equations with boundary conditions
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- A numerical method for solving systems of higher order linear functional differential equations
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- Nonlinear self adjointness, conservation laws and exact solutions of ill-posed Boussinesq equation
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- On combined optical solitons of the one-dimensional Schrödinger’s equation with time dependent coefficients
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- On soliton solutions of the Wu-Zhang system
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- Comparison between the (G’/G) - expansion method and the modified extended tanh method
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- On the union of graded prime ideals
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- Oscillation criteria for nonlinear fractional differential equation with damping term
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- A new method for computing the reliability of consecutive k-out-of-n:F systems
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- A time-delay equation: well-posedness to optimal control
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- Numerical solutions of multi-order fractional differential equations by Boubaker polynomials
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- Laplace homotopy perturbation method for Burgers equation with space- and time-fractional order
- Topical Issue: Recent Developments in Applied and Engineering Mathematics
- The calculation of the optical gap energy of ZnXO (X = Bi, Sn and Fe)
- Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
- Analysis of time-fractional hunter-saxton equation: a model of neumatic liquid crystal
- Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
- A certain sequence of functions involving the Aleph function
- Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
- On negacyclic codes over the ring ℤp + uℤp + . . . + uk + 1 ℤp
- Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
- Solitary and compacton solutions of fractional KdV-like equations
- Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
- Regarding on the exact solutions for the nonlinear fractional differential equations
- Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
- Non-local Integrals and Derivatives on Fractal Sets with Applications
- Special Issue: Advanced Computational Modelling of Nonlinear Physical Phenomena
- On the solutions of electrohydrodynamic flow with fractional differential equations by reproducing kernel method
- Special issue on Information Technology and Computational Physics
- On uninorms and nullnorms on direct product of bounded lattices
- Special issue on Information Technology and Computational Physics
- Phase-space description of the coherent state dynamics in a small one-dimensional system
- Special issue on Information Technology and Computational Physics
- Automated Program Design – an Example Solving a Weather Forecasting Problem
- Special issue on Information Technology and Computational Physics
- Stress - Strain Response of the Human Spine Intervertebral Disc As an Anisotropic Body. Mathematical Modeling and Computation
- Special issue on Information Technology and Computational Physics
- Numerical solution to the Complex 2D Helmholtz Equation based on Finite Volume Method with Impedance Boundary Conditions
- Special issue on Information Technology and Computational Physics
- Application of Genetic Algorithm and Particle Swarm Optimization techniques for improved image steganography systems
- Special issue on Information Technology and Computational Physics
- Intelligent Chatter Bot for Regulation Search
- Special issue on Information Technology and Computational Physics
- Modeling and optimization of Quality of Service routing in Mobile Ad hoc Networks
- Special issue on Information Technology and Computational Physics
- Resource management for server virtualization under the limitations of recovery time objective
- Special issue on Information Technology and Computational Physics
- MODY – calculation of ordered structures by symmetry-adapted functions
- Special issue on Information Technology and Computational Physics
- Survey of Object-Based Data Reduction Techniques in Observational Astronomy
- Special issue on Information Technology and Computational Physics
- Optimization of the prediction of second refined wavelet coefficients in electron structure calculations
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Droplet spreading and permeating on the hybrid-wettability porous substrates: a lattice Boltzmann method study
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- POD-Galerkin Model for Incompressible Single-Phase Flow in Porous Media
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Effect of the Pore Size Distribution on the Displacement Efficiency of Multiphase Flow in Porous Media
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Numerical heat transfer analysis of transcritical hydrocarbon fuel flow in a tube partially filled with porous media
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Experimental Investigation on Oil Enhancement Mechanism of Hot Water Injection in tight reservoirs
- Special Issue on Research Frontier on Molecular Reaction Dynamics
- Role of intramolecular hydrogen bonding in the excited-state intramolecular double proton transfer (ESIDPT) of calix[4]arene: A TDDFT study
- Special Issue on Research Frontier on Molecular Reaction Dynamics
- Hydrogen-bonding study of photoexcited 4-nitro-1,8-naphthalimide in hydrogen-donating solvents
- Special Issue on Research Frontier on Molecular Reaction Dynamics
- The Interaction between Graphene and Oxygen Atom
- Special Issue on Research Frontier on Molecular Reaction Dynamics
- Kinetics of the austenitization in the Fe-Mo-C ternary alloys during continuous heating
- Special Issue: Functional Advanced and Nanomaterials
- Colloidal synthesis of Culn0.75Ga0.25Se2 nanoparticles and their photovoltaic performance
- Special Issue: Functional Advanced and Nanomaterials
- Positioning and aligning CNTs by external magnetic field to assist localised epoxy cure
- Special Issue: Functional Advanced and Nanomaterials
- Quasi-planar elemental clusters in pair interactions approximation
- Special Issue: Functional Advanced and Nanomaterials
- Variable Viscosity Effects on Time Dependent Magnetic Nanofluid Flow past a Stretchable Rotating Plate