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Existence Theory and Numerical Solutions of the Fractional Order Mathematical Models

Thesis Info

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Author

Haq, Fazal

Program

PhD

Institute

Hazara University

City

Mansehra

Province

KPK

Country

Pakistan

Thesis Completing Year

2018

Thesis Completion Status

Completed

Subject

Physical Sciences

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/10861/1/Fazal%20Haq_Maths_2018_Hazara_PRR.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676724700284

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In last few decades, it has been proved that fractional order differential equations and their systems are very important in mathematical modeling various phenomena of biological, chemical and physical sciences. In addition to these, a fractional calculus also contains many applications in various fields of engineering and technology. For this propose , differential equations of fractional order is the point of attention in last few years. This project is related with the study of existence theory and numerical solutions of fractional order differential equations. For this study, we first review some useful definitions, notations and results from fractional calculus. Also for the study of numerical solutions, we use a power full techniques. We start our thesis with the study of existence and uniqueness of positive solutions for simple boundary value problem. Then, we obtain necessary and sufficient conditions for existence of at least three positive solutions for the considered models. To solve coupled systems of nonlinear fractional differential equations, we discussamethodwhichisknownasLaplaceAdomiandecompositionmethod(LADM). LADM is an excellent mathematical tool for solving linear and nonlinear differential equations. This method is a combination of the famous integral transform known as Laplace transform and the Adomain decomposition method. In this method, we handle some class of coupled systems of nonlinear fractional order differential equations. Using the proposed method to obtain successfully an exact or approximate solution in the form of convergence series. Thus, we can easily applyLADMto solveawideclassof nonlinearfractionalorderdifferentialequations.The suggested method is applied without any linearization, discretization and unrealistic assumptions. It has been proved that LADM is vary efficient and suitable to solve non-linear problem of physical nature. Some examples are presented to justify the accuracy and performance of the proposed method.
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محمد اسحق جلیس

محمد اسحق جلیس مرحوم
ابھی یہ سطور زیر تحریر ہی تھیں کہ مولوی محمد اسحق جلیس ندوی کے انتقال کی اطلاع ملی وہ ندوہ کے کتب خانہ کے نائب مہتمم، رسالہ تعمیر حیات کے ایڈیٹر اور تحریک پیام انسانیت کے سرگرم سکریٹری تھے، مولانا ابوالحسن علی ندوی ان کے بڑے قدرشناس تھے، اور ان سے بڑی توقعات رکھتے تھے، افسوس کہ عمر نے وفا نہ کی اور عین شباب کے عالم میں اس جہاں فانی سے رخصت ہوگئے، اﷲ تعالیٰ ان کو اپنی رحمت سے نوازے، پس ماندگان کو صبر عطا فرمائے اور ندوہ کو ان کا نعم البدل عطا فرمائے۔ (’’ع ، ق‘‘، جولائی ۱۹۷۹ء)

Which of Abrahim’s Son Was Sacrificed?

The issue of the sacrifice of Abrahim’s son remain an important bone of contention among the followers of Abrahamic religions. The Jews and Christians believe that it is Ishaq while among the Muslims some opine in favour of Ismail while favours Ishaq. To a common man it seems to be not an important issue whether it is Ismail or Ishaq but when its implications are taken into account then it is of great value to Muslims, Jews and Christians that who was sacrificed? Ismail or Ishaq. In the following pages an effort is made to look the issue in the light of some contemporary researches conducted in the Indo-Pak Subcontinent in the first half of the 20th century. Different positions on the issue are evaluated with the Quran and other holy scriptures and in the contemporary western scholarships.

Bitrophic Effects of Artificial Diets of American Bollworm Helicoverpa Armigera H. on Different Biological Aspects of Bracon Hebetor Say

Helicoverpa armigera (Hubner) is a notorious pest of various fields and horticultural crops worldwide. Insecticides application ultimately responsible for resistance development in H. armigera, unsafe for the environment and natural enemies of insect pests. To withstand the pesticides resistance, biological control successfully helps in reducing pest population. Bracon hebetor is a cosmopolitan species that have the ability to parasitize a wide range of lepidopteran and few coleopteran hosts. As H. armigera fed on wide host range so, quality and quantity of food nutrients provided at the time of Bracon hebetor larval development is a crucial step. The need of the current research project is to focus on which best suited semi-synthetic artificial diet of H. armigera; B. hebetor developmental parameters are good and further, how it can effectively be used under the scenario of ecological conditions after getting reared on efficient host for the control of H. armigera. So we can utilize B. hebetor as an efficient biological control agent for H. armigera management. Study objectives were to screen out the best suited artificial diet of H. armigera for the rearing of B. hebetor. Investigation of B. hebetor life history traits on most efficient hosts from the screened host species for the effective rearing of B. hebetor. Determination of the effect of temperature, relative humidity regimes and host larval densities on the life history traits of B. hebetor parasitizing H. armigera. B. hebetor parasitization, reproduction, development, fitness related traits and life table parameters were observed on different pyralids, noctuids, gelechiids and curculionids under laboratory conditions (25±2°C and 70±5%) in order to find out the most suitable host for the mass rearing and inundative release purposes to control H. armigera. Results indicated that among four semi-synthetic diets (gram, rice, wheat and corn) of H. armigera, gram flour mediated diet was responsible for good biological and developmental parameters of B. hebetor under laboratory conditions. It can be successfully used for the rearing of H. armigera as a host for good parasitism and development of B. hebetor. Though, H. armigera itself was not an excellent host for the continuous rearing of parasitoid therefore, seven different hosts (six from Lepidoptera and one Coleoptera) were utilized to screen out the suitable one. Results showed that the pyralids hosts (E. kuehniella, C. cephalonica and G. mellonella) gave the best results for daily fecundity, parasitoid survival to adulthood, development time and female biased sex ratio followed by H. armigera and the worst reproduction occurred on H. postica. Among the most efficient screened pyralid hosts, B. hebetor biological parameters and fitness related traits were excellent on E. kuehniella as compared to C. cephalonica. So production of good fitness traits (wing area, tibia size and dry mass), excellent biological parameters (good immature numbers and adults emergence), short egg-adult development time and changes in THCs provide a good means for the immature development of B. hebetor within the parasitized E. kuehniella for several days. Temperature relative humidity regimes and H. armigera larval densities significantly affect the biological and developmental parameters of B. hebetor. Excellent life history parameters of B. hebetor parasitizing gram diet fed H. armigera were observed at 25°C and 60% R.H. combination. Among various larval densities of H. armigera good biological parameters of B. hebetor like production of immature numbers, short egg-adult development time, survivorship and female progeny production was significantly good on five larval density of H. armigera offered for parasitization. Therefore, it is recommended that gram mediated diet is best suited for the rearing of H. armigera in order to find out the best suited temperature, humidity combination and host larval density in the field release planning and maintenance of B. hebetor population for long term field crops management to control H. armigera. On the other hand B. hebetor quality progeny production on E. kuehniella make this insect of great potential to serve as valuable material for laboratory rearing and inundative field release to control H. armigera larval populations.