On Comparison Study between Double Sumudu and Elzaki Linear Transforms Method for Solving Fractional Partial Differential Equations

في هذا البحث، استخدمنا طريقتي التحويلان التكامليان المضاعفان سمودو و الزاكي للحصول على الحلول العددية لبعض انواع المعادلات التفاضلية الجزئية الكسرية ذات المعاملات الثابتة، ووضحنا كفاءة الطريقة بتقديم بعض الامثلة العددية المحسوبة باستخدام برنامجي ماثكاد 15 و ماتلاب R2015a        In this paper, double Sumudu and double Elzaki transforms methods are used to compute the numerical solutions for some types of fractional order partial differential equations with constant coefficients and explaining the efficiently of the method by illustrating some numerical examples that are computed by using  Mathcad 15.and graphic in Matlab R2015a

Numerical and analytic method for solvingproposal New Type for fuzzy nonlinear volterra integral equation

In this paper, we proved the existence and uniqueness and convergence of the solution of new type for nonlinear fuzzy volterra integral equation . The homotopy analysis method are proposed to solve the new type fuzzy nonlinear Volterra integral equation . We convert a fuzzy volterra integral equation for new type of kernel for integral equation, to a system of crisp function nonlinear volterra integral equation . We use the homotopy analysis method to find the approximate solution of the system and hence obtain an approximation for fuzzy solution of the nonlinear fuzzy volterra integral equation . Some numerical examples is given and results reveal that homotopy analysis method is very effective and compared with the exact solution and calculate the absolute error between the exact and AHM .Finally using the MAPLE program to solve our problem

Solvability of Some Types for Multi-fractional Integro-Partial Differential Equation

خلال البحث تم المناقشة وبالتفصيل قابلية الحل لبعض الأنظمة من المعادلات المتعددة التفاضلية التكاملية ذات الرتب الكسرية باستخدام مفهوم مسائل كوشي المختصرة وايضا نظرية شبه زمرة المختارة مع بعض الشروط الضرورية والكافية.In this article, the solvability of some proposal types of the multi-fractional integro-partial differential system has been discussed in details by using the concept of abstract Cauchy problem and certain semigroup operators and some necessary and sufficient conditions.

Convergence of the Generalized Homotopy Perturbation Method for Solving Fractional Order Integro-Differential Equations

In this paper,the homtopy perturbation method (HPM) was applied to obtain the approximate solutions of the fractional order integro-differential equations . The fractional order derivatives and fractional order integral are described in the Caputo and Riemann-Liouville sense respectively. We can easily obtain the solution from convergent the infinite series of HPM . A theorem for convergence and error estimates of the HPM for solving fractional order integro-differential equations was given. Moreover, numerical results show that our theoretical analysis are accurate and the HPM can be considered as a powerful method for solving fractional order integro-diffrential equations

The Necessary and Sufficient Optimality Conditions for a System of FOCPs with Caputo–Katugampola Derivatives

The necessary optimality conditions with Lagrange multipliers  are studied and derived for a new class that includes the system of Caputo–Katugampola fractional derivatives to the optimal control problems with considering the end time free. The formula for the integral by parts has been proven for the left Caputo–Katugampola fractional derivative that contributes to the finding and deriving the necessary optimality conditions. Also, three special cases are obtained, including the study of the necessary optimality conditions when both the final time  and the final state  are fixed. According to convexity assumptions prove that necessary optimality conditions are sufficient optimality conditions

Holder's Inequality ρ–Mean Continuity for Existence and Uniqueness Solution of Fractional Multi-Integrodifferential Delay System

We herein present the detailed results for the existence and uniqueness of mild solution for multifractional order impulsive integrodifferential control equations with a nonlocal condition involving several types of semigroups of bounded linear operators, which were established on probability density functions related with the fractional differential equation. Additionally, we present the necessary and sufficient conditions to investigate Schauder’s fixed point theorem with Holder’s inequality ρ–mean continuity and infinite delay parameter to guarantee the uniqueness of a fixed point

Solution of Some Types for Composition Fractional Order Differential Equations Corresponding to Optimal Control Problems

The approximate solution for solving a class of composition fractional order optimal control problems (FOCPs) is suggested and studied in detail. However, the properties of Caputo and Riemann-Liouville derivatives are also given with complete details on Chebyshev approximation function to approximate the solution of fractional differential equation with different approach. Also, the relation between Caputo and Riemann-Liouville of fractional derivative took a big role for simplifying the fractional differential equation that represents the constraints of optimal control problems. The approximate solutions are defined on interval [0,1] and are compared with the exact solution of order one which is an important condition to support the working method. Finally, illustrative examples are included to confirm the efficiency and accuracy of the proposed method

SOME NEW RELATIONSHIPS BETWEEN THE FRACTIONAL DERIVATIVES OF FIRST, SECOND, THIRD AND FOURTH CHEBYSHEV WAVELETS

Soil acidity and salinity are major constraints for agricultural productivity around the world. Improving beneficial plant-microbe interactions in the rhizosphere and endosphere could improve plant tolerance to acidity and salinity and increase crop yield and productivity. However, soil acidity and salinity can adversely impact symbiotic interactions of plants with arbuscular mycorrhizal fungi (AMF) and nitrogen-fixing bacteria (NFB), which play an important role in plant nutrition. Soil amendments or foliar application of signaling compounds were shown to modulate microbial diversity and composition in the rhizosphere and plant endosphere and could be potential management tools to improve beneficial plant-microbial interactions in acidic and saline soils. However, it is not clear how native microbial community in the rhizosphere and endosphere of a legume crop respond to acidity and salinity stress and soil amendments. It is also not clear whether exogenous application of signaling compounds can significantly modulate rhizosphere and endosphere microbiome structure of a legume crop. Particularly, AMF and NFB responses to soil amendments and signaling compounds needs to be investigated in acidic and saline soils. To address these knowledge gaps, two studies were conducted to evaluate soil and foliar amendments to improve plant-microbe interactions in a legume crop (cowpeas, Vigna unguiculata (L.) Walp.) grown in an acidic and a saline soil. A comprehensive assessment of plant growth and development and microbiome composition in the rhizosphere and endosphere including legume-rhizobia and plant root-AMF symbiosis were evaluated in both studies. In the first experiment, biochar as a soil amendment and salicylic acid (SA) as a foliar applied stimulant were evaluated for their impacts on soil pH, nutrient uptake, nodulation, AMF colonization, diversity and composition of rhizosphere and endophytic microbiome of cowpea plants grown in acidic soils (pH = 4.8). Results showed that soil acidity reduced nodulation, plant nutrient concentrations, diversity of rhizosphere microbes and pod yield. Biochar (BC) amendment was more effective in improving plant nutrient uptake and pod yields than SA treatment. Soil pH was increased to around 5.8 ± 0.2 in the BC treatment compared to control (5.0 ± 0.2). Similarly, nodulation numbers were higher in BC treatment, which resulted in higher N concentrations in the leaves compared to SA treatment. Percent AMF colonization was also increased significantly in BC treatment, which recorded higher leaf P concentrations. Treatment of SA significantly improved AMF colonization and abundance of AMF taxa in the rhizosphere, however, plant nutrient concentrations and pod yield did not significantly differ from unamended control. Both BC and SA significantly altered the microbial composition in the rhizosphere and plant endosphere. Treatment of BC also significantly increased the relative abundance of several plant beneficial taxa such as Bacillus, Pseudomonas, Penicillium, Rhizobium and Bradyrhizobium compared to control. Based on the results of this study it was concluded that BC application to an acidic soil was effective in improving plant-microbe interactions and pod yields of cowpea plants grown in an acidic soil. A second greenhouse study was conducted to evaluate compost (CMP) and gypsum (GYP) as soil amendments and foliar application of strigolactones (SL), salicylic acid (SA) and coumarins (COU) for their impacts on plant-microbe interactions of cowpea grown in a saline soil (pH = 8.5). Crop growth and development, diversity and composition of rhizosphere and endophytic microbiome, arbuscular mycorrhizal fungi (AMF) colonization, nodulation, plant nutrient concentrations were estimated from the treatments. Results showed that soil salinity adversely impacted plant nutrient uptake, AMF colonization and pod yields. Among the amendment treatments, SL+SA treatment produced the highest cowpea pod yield followed by CMP amendment. Highest nodulation and root colonization were noted in SL+SA treated plants. Significantly higher relative abundance of several plant beneficial microbes including a Streptomyces species and several AMF (Rhizophagus and Diversispora) were noted in the rhizosphere and roots of SL+SA treated plants compared to control. Based on the results of this study, it was concluded that foliar application of SL+ SA was most effective in improving plant-microbe interactions and pod yield of cowpea plant grown in a saline soil. It was demonstrated by these two studies that several beneficial microbes in the rhizosphere and endosphere of a legume crop were sensitive to acidity and salinity stress. It was also clear that various soil amendments and exogenous application of signaling compounds significantly altered rhizosphere and endosphere microbiome structure of a legume crop, and improved cowpea interactions with AMF and NFB. Using effective soil amendments such as biochar in acidic soil and foliar application of SL and SA for plants grown in saline soils are potential agriculture management avenues for improving soil health and productivity in acidic and saline soils