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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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Study of Electrical and Magnetic Properties of Strontium-Barium Hexaferrite Nanomaterials for Potential Technological Applications

In recent years, there has been intense search for materials which are ferri- magnetic and possess high values of electrical resistivity and magnetization together with low dielectric losses. Such materials are considered to have potential for useful applications in technology. Cationic doping in ferrites presents a promising technique for achieving such characteristics in ferrites. Here we report that doping hexagonal strontium-barium nanoferrites (Sr 0.5 Ba 0.5 Fe 12 O 19 ) with a binary mixture of RE-Ni (RE = La, Ce, Pr and Nd) can surprisingly modify the properties of hitherto undoped strontium-barium hexaferrite. Different series of nanosized strontium-barium hexaferrite of nominal composition of Sr 0.5 Ba 0.5-x RE x Fe 12-y N iy O 19 (x = 0.00-0.10; y = 0.00-1.00) are prepared by the chemical co-precipitation method. The formation of nanosized hexaferrites with single hexagonal phase and homogenous morphology is probed using transmission electron microscopy, X-ray diffraction and scanning electron microscopic analyses. The electrical properties are measured by the DC- electrical resistivity and the dielectric measurements. The magnetic susceptibility is measured by a magnetic susceptometer and the hysteresis loops are traced by the DC- magnetization measurements. The results obtained reveal that, by introducing a relatively small amount of RE-Ni (RE = La, Ce, Pr and Nd) ions, an important modification of the structural, electrical and magnetic properties can be obtained. Thermal analysis reveals magnetoplumbite phase begins to be formed at ~1000 K and is completed at 1323 K; a conclusion also complimented by XRD studies. XRD studies show that all the synthesized samples are composed of a single magnetoplumbite phase. The average crystallite sizes in differently doped samples are in the range of 18-48 nm. TEM analysis of the undoped strontium-barium hexaferrite shows that particles appear to have well-defined hexagonal geometry having an average particle size in the range of 30-40 nm, which is comparable to the size of 36 nm calculated by the Scherrer equation. The results of SEM analysis indicate the appearance of homogenous surface of the synthesized sample with uniform particle sizes. Temperature dependence of electrical resistivity reflects the semi-conducting nature of the ferrites. The values of electrical resistivity and activation energy increase by substitution with different dopant contents of La-Ni, Ce-Ni and Pr-Ni up to a certain level but decrease by substitution with Nd-Ni. The dielectric constant and dielectric loss tangent decrease by increasing applied field frequency. The values ofdrift mobility, dielectric constant and loss tangent decrease by increasing the La-Ni, Ce-Ni and Pr-Ni contents but they increase by increasing Nd-Ni content. Possible mechanisms for the above mentioned result are discussed in terms of conduction mechanism via electron-hopping between Fe 2+ and Fe 3+ ions and Maxwell-Wagner type of interfacial polarization. The value of Curie temperature decreases by the substitution of RE-Ni (La, Ce, Pr and Nd) ions, which has been explained in terms of the strength of exchange interactions. Doping with Ce-Ni, Pr-Ni and Nd-Ni, increases both the saturation magnetization and remanence in the ranges 66-102 emu/g and 43- 64 emu/g, respectively, but substitution with La-Ni ions results in their decrease. However, the value of coercivity decreases by increasing the content of La-Ni, Ce-Ni and Pr-Ni ions. The variations in the magnetic parameters shown by Mossbauer spectral analysis are explained on the basis of preferential site occupancy of the substituted cations. The compositional variation of hyperfine interaction parameters confirms the site occupancy of the doped cations in hexaferrite materials. The value of isomer shift for 2b, 12k and 4f 2 sites decreases by increasing the content of RE-Ni (RE = La, Ce, Pr and Nd) ions. The value of quadrupole splitting for 2b site in the undoped strontium-barium hexaferrite decreases by the substitution of rare-earth ions and it slightly increases for the 4f 2 site. The variation of hyperfine magnetic field with dopant content of RE-Ni ions is akin to the compositional variation of saturation magnetization of RE-Ni doped strontium-barium hexaferrites. The results of the study have shown potential for application of these materials in the area of magnetic recording, permanent magnetic materials and surface mount devices for fabricating multilayer chip inductors. The Nd-Ni doped strontium-barium hexaferrites may be more useful as permanent magnets, owing to high values of magnetization and coercivity. High electrical resistivity and low dielectric losses are the characteristics required to reduce the eddy current losses.