A study of the GAM approach to solve laminar boundary layer equations in the presence of a wedge

We apply an easy and simple technique, the generalized ap- proximation method (GAM) to investigate the temperature field associated with the Falkner-Skan boundary-layer problem. The nonlinear partial differ- ential equations are transformed to nonlinear ordinary differential equations using the similarity transformations. An iterative scheme for the non-linear ordinary differential equations associated with the velocity and temperature profiles are developed via GAM. Numerical results for the dimensionless ve- locity and temperature profiles of the wedge flow are presented graphically for different values of the wedge angle and Prandtl number

Positive Solutions to Nonlinear Higher-Order Nonlocal Boundary Value Problems for Fractional Differential Equations

We study existence of positive solutions to nonlinear higher-order nonlocal boundary value problems corresponding to fractional differential equation of the type 0+()+(,())=0, ∈(0,1), 0<<1. (1)=()+2, (0)=()−1, (0)=0, (0)=0⋯(−1)(0)=0, where, −1<<, (≥3)∈ℕ, 0<,,<1, the boundary parameters 1,2∈ℝ+ and 0+ is the Caputo fractional derivative. We use the classical tools from functional analysis to obtain sufficient conditions for the existence and uniqueness of positive solutions to the boundary value problems. We also obtain conditions for the nonexistence of positive solutions to the problem. We include examples to show the applicability of our results

Phytochemical and cytotoxic analysis of Pharthenium hysterophosis selected from District Bannu, Pakistan

Parthenium hysterophorus is a well known medicinal plant widely used traditionally in the treatment of various diseases and as a constituent of various drugs, and in phytotherapy. The current study was designed to investigate the phytochemical screening and cytotoxic capacity of methanolic and n-hexane extract of P. hysterophorus. Quantitative analysis of P. hysterophorus showed maximum quantity of flavonoids in methanolic extract of P. hysterophorus which turned down gradually in n-hexane extract of P. hysterophorus due to the decrease in organic solvents polarity. Similar results were also observed for saponins and tannins during this investigation. The highest quantity of alkaloids was recorded in the methanolic extract of P. hysterophorus when compared to n-hexane extract. The extracts also showed maximum cytotoxic potential in various concentrations of n-hexane and methanolic extract of P. hysterophorus. The results revealed that P. hysterophorus contains a remarkable cytotoxic activity due to the presence of bioactive constituents.Key words: Parthenium hysterophorus, cytotoxic, phytochemical screening

Effects of different doses of fertilizers on growth and yield components of biodiesel plant (Jatropha curcas L.)

Effects of different doses of fertilizers on growth and yield components of biodiesel plant (Jatropha curcas L.), was made in the prevailing environment of University of Science and Technology Bannu. The experiment was carried out in randomized complete block design (RCBD) with three replications. Nitrogen, phosphorus and potassium were applied in the form of urea, single super phosphate and potash in the shape of combined NPK, respectively, during sprouting of leaves and thereafter with each irrigation in split doses. The analysis of variance for number of branches per plant showed significant results indicated in ANOVA. Significantly the number of primary branches per plant 8.07 was recorded in T4 with the application of dose of (N P K: 2507.5 gm: 1091.5 gm: 501.5 gm). Maximum No. of secondary branches per plant (9.100) were recorded in T1 especially at both plots. Flower per inflorescence showed that maximum value (122.810) was recorded in T5 of especially at both plots. Inflorescence plant-1 showed that maximum value (39.477) were recorded in T1 and minimum value were noted as (12.733) at T3 on both the flats. The data for fruits per plant showed that maximum value (143.300) was recorded in T1. The seed index, which was the output of the aim of this research work, reflects highly significant differences in mean value observed for both subplots. Mean values showed that maximum seeds weight 45.86 was recorded in T1 of especially at both plots while low seed index was noted as 22.84 at T5 (N: 3400 gm, P: 1480 gm, K: 680 gm) at both the subplots

Existence and approximation of solutions of nonlinear boundary value problems

In chapter two, we establish new results for periodic solutions of some second order non-linear boundary value problems. We develop the upper and lower solutions method to show existence of solutions in the closed set defined by the well ordered lower and upper solutions. We develop the method of quasilinearization to approximate our problem by a sequence of solutions of linear problems that converges to the solution of the original problem quadratically. Finally, to show the applicability of our technique, we apply the theoretical results to a medical problem namely, a biomathematical model of blood flow in an intracranial aneurysm. In chapter three we study some nonlinear boundary value problems with nonlinear nonlocal three-point boundary conditions. We develop the method of upper and lower solutions to establish existence results. We show that our results hold for a wide range of nonlinear problems. We develop the method of quasilinearization and show that there exist monotone sequences of solutions of linear problems that converges to the unique solution of the nonlinear problems. We show that the sequences converge quadratically to the solutions of the problem in the C1 norm. We generalize the technique by introducing an auxiliary function to allow weaker hypotheses on the nonlinearity involved in the differential equations. In chapter four, we extend the results of chapter three to nonlinear problems with linear four point boundary conditions. We generalize previously existence results studied with constant lower and upper solutions. We show by an example that our results are more general. We develop the method of quasilinearization and its generalization for the four point problems which to the best of our knowledge is the first time the method has been applied to such problems. In chapter five, we extend the results to second order problems with nonlinear integral boundary conditions in two separate cases. In the first case we study the upper and lower solutions method and the generalized method of quasilinearization for the Integral boundary value problem with the nonlinearity independent of the derivative. While in the second case we show the nonlinearity to depend also on the first derivative. Finally, in chapter six, we study multiplicity results for three point nonlinear boundary value problems. We use the method of upper and lower solutions and degree arguments to show the existence of at least two solutions for certain range of a parameter r and no solution for other range of the parameter. We show by an example that our results are more general than the results studied previously. We also study existence of at least three solutions in the pressure of two lower and two upper solutions for some three-point boundary value problems. In one problem, we employ a condition weaker than the well known Nagumo condition which allows the nonlinearity f(t, x, x’) to grow faster than quadratically with respect to x’ in some cases

The generalized approximation method and nonlinear heat transfer equations

Generalized approximation technique for a solution of one-dimensional steady state heat transfer problem in a slab made of a material with temperature dependent thermal conductivity, is developed. The results obtained by the generalized approximation method (GAM) are compared with those studied via homotopy perturbation method (HPM). For this problem, the results obtained by the GAM are more accurate as compared to the HPM. Moreover, our (GAM) generate a sequence of solutions of linear problems that converges monotonically and rapidly to a solution of the original nonlinear problem. Each approximate solution is obtained as the solution of a linear problem. We present numerical simulations to illustrate and confirm the theoretical results

In vitro pharmacological effects of Astragalus eremophilus and Melilotus parviflora

Traditional medicines are composed of herbal formulations and their active ingredients and constituents which play a crucial role in the treatment of various human ailments. Astragalus eremophilus and Melilotus indicus (L.) All. (syn. Melilotus parviflora Desf.) are used traditionally as antiperspirant, tonic, diuretic, laxative and narcotic agents. The current study was designed to investigate the Astragalus eremophilus and Melilotus indicus (L.) All. (syn. Melilotus parviflora Desf.) methanol extracts for their antioxidant, antibacterial and antifungal activities. Fine powder of A. eremophilus and M. parviflora was extracted with 70% methanol to get crude methanol extract. Extract was characterized for antioxidant, antibacterial and antifungal activities. Antioxidant activity of various concentrations (3 mg/ml, 1.5 mg/ ml, 0.75 mg/ml, and 0.38 mg/ml) of both plant extracts was analyzed using 2,2-Diphenyl-1-picrylhydrazyl (DPPH) free radical. Salmonella typhemorium, Klebsiella pneumoniae (gram-negative) and Staphylococcus aureus, Enterococcus faecalis (gram-positive) bacterial strains were used for assessment of antibacterial activities. Antifungal activities of 7.5 mg/ml, 5.0 mg/ml, 2.5 mg/ml (A. eremophilus and M. parviflora) were conducted using Aspergillus niger, Aspergillus flavus, Aspergillus fumigatus and Candida albicons. At high concentration (3 mg/ml), all the tested fractions of A. eremophilus and M. parviflora methanol extracts showed potent antioxidant activities, ranging between 83.8 and 63.33%. Antibacterial activities revealed that A. eremophilus showed a maximum zone of inhibition (8.1 ± 0.1) on Salmonella typhenorium followed by Enterococcus faecalis (7.2 ± 0.1), Klebsellesa pneumonia (6.1 ± 0.6), and Staphylococcus aureus (5.1 ± 0.4), and at highest concentration (7.5 mg/ml), however, maximum zone of inhibition of Melilotus parviflora was at 7.5 mg/ml followed by 5.0 mg/ml and 2.5 mg/ml against Klebsiella pneumonia, Staphylococcus aureus, Salmonella typhemorium and Enterococcus faecalis. Antifungal assessment of both plant extracts showed that the higher concentration (7.5 mg/ml) has significant inhibitory effect as compared to control. The results can lead to the conclusion that A. eremophilus and M. parviflora methanol extracts are indeed sources of potential therapeutic compounds against antibacterial, antifungal and free radical associated disorders