The analysis of Homotopy Perturbation Method (HPM) for the solution of fractional partial differential equations (FPDEs) is presented. A unified convergence theorem is given. In order to validate the theory, the solution of fractional-order Burger-Poisson (FBP) equation is obtained. Furthermore, this work presents the method to find the solution of FPDEs, while the same partial differential equation (PDE) with ordinary derivative i.e., for $\alpha =1$ , is not defined in the given domain. Moreover, HPM is applied to a complicated obstacle boundary value problem (BVP) of fractional order.

An efficient regeneration protocol was applied to regenerate shoots on salt stress-tolerant calli lines of aubergine (Solanum melongena). These NaCl-tolerant cell lines were obtained by two different methods. On the one hand, the developed callus tissue was transferred to a medium with a continuous salt content of 40, 80, 120, or 160 mM NaCl. On the other hand, the callus tissue was subjected to a stepwise increasing salinity to 160 mM NaCl every 30 days. With the second method, calli which could be selected were characterized by compact growth, a greenish color, and absence of necrotic zones. When grown on salt-free medium again, NaCl-tolerant calli showed a decline in relative growth rate and water content in comparison to the control line. This was more obvious in the 120 mM NaCl-tolerant callus. Lipid peroxidase activity increased in 40 and 80 mM NaCl-tolerant calli; yet did not increase further in 120 mM-tolerant callus. An increase in ascorbic acid content was observed in 80 and 120 mM NaCl-tolerant calli compared to the 40 mM NaCl-tolerant lines, in which ascorbic acid content was twice that of the control. All NaCl-tolerant lines showed significantly higher superoxide dismutase (SOD) (208–305–370 µmol min⁻¹ mg⁻¹ FW) and catalase (CAT) (136–211–238 µmol min⁻¹ mg⁻¹ FW) activities compared to control plants (231 and 126 µmol min⁻¹ mg⁻¹ FW). Plants were regenerated on the calli lines that could tolerate up to 120 mM NaCl. From the 32 plants tested in vitro, ten plants with a higher number of leaves and root length could be selected for further evaluation in the field. Their high salt tolerance was evident by their more elevated fresh and dry weight, their more increased relative water content, and a higher number and weight of fruits compared to the wild-type parental control. The presented work shows that somaclonal variation can be efficiently used to develop salt-tolerant mutants.

Previously, an efficient regeneration protocol was established and applied to regenerate plants from calli lines that could grow on eggplant leaf explants after a stepwise in vitro selection for tolerance to salt stress. Plants were regenerated from calli lines that could tolerate up to 120 mM NaCl. For further in vitro and in vivo evaluation, four plants with a higher number of leaves and longer roots were selected from the 32 plants tested in vitro. The aim of this study was to confirm the stability of salt tolerance in the progeny of these four mutants (‘R18’, ‘R19’, ‘R23’ and ‘R30’). After three years of in vivo culture, we evaluated the impact of NaCl stress on agronomic, physiological and biochemical parameters compared to the parental control (‘P’). The regenerated and control plants were assessed under in vitro and in vivo conditions and were subjected to 0, 40, 80 and 160 mM of NaCl. Our results show significant variation in salinity tolerance among regenerated and control plants, indicating the superiority of four regenerants (‘R18’, ‘R19’, ‘R23’ and ‘R30’) when compared to the parental line (‘P’). In vitro germination kinetics and young seedling growth divided the lines into a sensitive and a tolerant group. ‘P’ tolerate only moderate salt stress, up to 40 mM NaCl, while the tolerance level of ‘R18’, ‘R19’, ‘R23’ and ‘R30’ was up to 80 mM NaCl. The quantum yield of PSII (Φ_PSII) declined significantly in ‘P’ under salt stress. The photochemical quenching was reduced while nonphotochemical quenching rose in ‘P’ under salt stress. Interestingly, the regenerants (‘R18’, ‘R19’, ‘R23’ and ‘R30’) exhibited high apparent salt tolerance by maintaining quite stable Chl fluorescence parameters. Rising NaCl concentration led to a substantial increase in foliar proline, malondialdehyde and soluble carbohydrates accumulation in ‘P’. On the contrary, ‘R18’, ‘R19’, ‘R23’ and ‘R30’ exhibited a decline in soluble carbohydrates and a significant enhancement in starch under salinity conditions. The water status reflected by midday leaf water potential (ψl) and leaf osmotic potential (ψπ) was significantly affected in ‘P’ and was maintained a stable level in ‘R18’, ‘R19’, ‘R23’ and ‘R30’ under salt stress. The increase in foliar Na⁺ and Cl⁻ content was more accentuated in parental plants than in regenerated plants. The leaf K⁺, Ca²⁺ and Mg²⁺ content reduction was more aggravated under salt stress in ‘P’. Under increased salt concentration, ‘R18’, ‘R19’, ‘R23’ and ‘R30’ associate lower foliar Na⁺ content with a higher plant tolerance index (PTI), thus maintaining a normal growth, while foliar Na⁺ accumulation was more pronounced in ‘P’, revealing their failure in maintaining normal growth under salinity stress. ‘R18’, ‘R19’, ‘R23’ and ‘R30’ showed an obvious salt tolerance by maintaining significantly high chlorophyll content. In ‘R18’, ‘R19’, ‘R23’ and ‘R30’, the enzyme scavenging machinery was more performant in the roots compared to the leaves. Salt stress led to a significant augmentation of catalase, ascorbate peroxidase and guaiacol peroxidase activities in the roots of ‘R18’, ‘R19’, ‘R23’ and ‘R30’. In contrast, enzyme activities were less enhanced in ‘P’, indicating lower efficiency to cope with oxidative stress than in ‘R18’, ‘R19’, ‘R23’ and ‘R30’. ACC deaminase activity was significantly higher in ‘R18’, ‘R19’, ‘R23’ and ‘R30’ than in ‘P’. The present study suggests that regenerated plants ‘R18’, ‘R19’, ‘R23’ and ‘R30’ showed an evident stability in tolerating salinity, which shows their potential to be adopted as interesting selected mutants, providing the desired salt tolerance trait in eggplant.

In this paper, we have analyzed the first- and second-order operator splitting schemes for the numerical solution of the Allen-Cahn equation. Different possibilities are considered for the derivation of the schemes. The numerical errors are computed in order to verify the effectiveness of each scheme. Validation of the schemes is provided by numerically solving some well-known examples.