Solving Boundary Value Problems for Ordinary Differential Equations Using Direct Integration and Shooting Techniques

In this thesis, an efficient algorithm and a code BVPDI is developed for solving Boundary Value Problems (BVPs) for Ordinary Differential Equations (ODEs). A generalised variable order variable stepsize Direct Integration (01) method, a generalised Backward Differentiation method (BDF) and shooting techniques are used to solve the given BVP. When using simple shooting technique, sometimes stability difficulties arise when the differential operator of the given ODE contains rapidly growing and decaying fundamental solution modes. Then the initial value solution is very sensitive to small changes in the initial condition. In order to decrease the bound of this error, the size of domains over which the Initial Value Problems (IVPs) are integrated has to be restricted. This leads to the multiple shooting technique, which is generalisation of the simple shooting technique. Multiple shooting technique for higher order ODEs with automatic partitioning is designed and successfully implemented in the code BVPDI, to solve the underlying IVP. The well conditioning of a higher order BVP is shown to be related to bounding quantities, one involving the boundary conditions and the other involving the Green's function. It is also shown that the conditioning of the multiple shooting matrix is related to the given BVP. The numerical results are then compared with the only existing direct method code COLNEW. The advantages in computational time and the accuracy of the computed solution, especially, when the range of interval is large, are pointed out. Also the advantages of BVPDI are clearer when the results are compared with the NAG subroutine D02SAF (reduction method). Stiffness tests for the system of first order ODEs and the techniques of identifying the equations causing stiffness in a system a rediscussed. The analysis is extended for the higher order ODEs. Numerical results are discussed indicating the advantages of BVPDI code over COLNEW. The success of the BVP DI code applied to the general class of BVPs is the motivation to con sider the same code for a special class of second order BVPs called Sturm-Liouville (SL) problems. By the application of Floquet theory and shooting algorithm, eigenvalues of SL problems with periodic boundary conditions are determined without reducing to the first order system of equations. Some numerical examples are given to illustrate the success of the method. The results are then compared, when the same problem is reduced to the first order system of equations and the advantages are indicated. The code BVPDI developed in this thesis clearly demonstrates the efficiency of using DI Method and shooting techniques for solving higher order BVP for ODEs

A Toxicity study on Kirubakara Shanmuga Chendhuram

Kirubakara shanmuga chenduram is being widely used in siddha medicine prepared according to the process found in Pathartha Guna Vilakkam (thaathu vagupu) (page. no. 55-56).It contains ingredients of Vaalai rasam, Gandhagam, Lingam, Veeram, Pooram, Manosilai, Thalagam, Uppu and Pirandai and cures diseases like Eight types of Thadippu, Kuttam, Megaranam, Gunmam, Kaasam, Sanni, Suram,Utkuthu, Purakuthu, Soolai, Moorchai and Vatha disease. The aim of the dissertation is to study the acute and 28 days repeated oral toxicity of the drug Kirubakara shanmuga chenduram in experimental animals. The qualitative analysis of the drug Kirubakara shanmuga chenduram contains the presence of Calcium, Ferrous Iron, Sulphate, Flouride and Oxalates. The quantitative analysis of Kirubakara shanmuga chenduram by ICP- OES contains the presence of mercury in ppm level and arsenic ,cadmium, lead were present in below detectable limit showed in Table A. The elemental analysis of XRF study showed in Table B. In this heavy element arsenic was found in ppm level. Though the major ingredients of the drug KSC are mercury, mecurial compounds and arsenic compounds, in the final product the mercury and arsenic were found in the ppm level. In HR- SEM analysis the particle size of Kirubakara Shanmuga Chenduram was 50 to 100 nm, mixed (Rectangle and Spherical) in shape. Surface was found smooth so the flowability was normal. The pH level is 8.9 to 9.1. It reveals the foremost preparation of the medicine KSC was done as per literature. The FTIR spectrum of graph shows alcohols, phenols, carboxylic acids, alkanes, alkynes, 1° amines, aromatics, esters, ethers, alkyl halides, aliphatic amines, alkenes, disulfides present in Kirubakara Shanmuga Chenduram. The prepared medicine Kirubakara Shanmuga Chenduram was subjected to acute and repeated oral toxicity studies in wistar albino rats. During acute toxicity study the animals were fasted overnight given single oral dose of 2000 mg/kg/body weight of Kirubakara Shanmuga Chenduram suspended in 10% aqueous solution of tween 18 and the animals were observed for 14 days. After first 4 hrs of drug administration all the three animals showed the following characteristic activities like alertness, grooming, touch response listlessness, unching, porphyrin, motor incordination, muscular spasm in all the three animals. After 24 hrs 2 animals were found dead and 48 hrs third animal was also found dead. 5, 50, 300 mg/kg/body weight displayed no signs of toxicity. Alertness, grooming, touch response were present. The acute toxicity study of KSC reported the LD 50 value of 500 mg/kg/body weight as per OECD guidelines. In repeated oral toxicity study, the doses selected for the study were x, 5x, 10x mg/kg body weight. During 28 days study there was no signs of toxic effect in all the animals and the body weight was gained significantly as that of control. The average water and food intake in the control group was not significantly (p>0.05) different from that of the test groups. the feed conversion efficiency followed the same pattern, thus indicating the normal metabolism of the animals. At the end of 28 th day all animals were sacrificed and the gross necropsy does not show any abnormality in the organs. In hematological parameter the levels of RBC and hemoglobin in experimental animals were elevated when compared with that of the control. It is statistically significant (p<0.05). On the other hand, there was a significant increase (p<0.05) in the ALP, at mid dose (5x) and high dose (10x) dose treated groups when compared to the control groups. Other hematological and biochemical investigations the alterations in values was obtained within normal biological and laboratory limits or the effect was not dose dependent. In the histopathology study of KSC administered to low dose (x), mid dose (5x), high dose (10x), the sections of heart, lungs, liver, kidney, brain were found normal as comparable with that of the control. Sections of spleen showed normal appearing white pulp with congested red pulp

Computing eigenvalues of periodic Sturm-Liouville problems using shooting technique and direct integration method

This paper deals with computing eigenvalues of periodic Sturm-Liouville (SL) problem by direct integration (DI) method using shooting technique without reducing to the system of first order ODEs. Floquet theory is applied to find a non-trivial solution of SL problems and the eigenvalues are approximated by the application of shooting techniques. Computational advantages are presented comparing the results obtained by the DI method with that of reducing to the system of first order ODEs

Enhanced Dwarf Mongoose Optimization Based Node Localization Scheme for Underwater Wireless Sensor Networks

Recently, the Underwater Wireless Sensor Network (UWSN) with movable nodes has been launched because it has a lot of potential uses in aquatic science and is getting a lot of attention from researchers. Because there are so many more possible underwater uses, it is important to make sure that signals can get from one underwater point to another. Because of the long lags in signal transmission and the changing speed of sound underwater, it is hard and not possible to use the usual localization method in UWSN. Node localization (NL) in UWSN tries to figure out where new nodes are by using known nodes. The correctness of location can have a big effect on how well a UWSN works. UWSN has a lot of trouble with precise NL. When it comes to WSN, NL problems mean figuring out where unknown sensor nodes (SNs) are. This shows how important it is to have a perfect NL system. The Enhanced Dwarf Mongoose Optimization based Node Localization Scheme (EDMO-NLS) for UWSN is being worked on in this study. The scout group, the alpha group, and the babysitters are the three social groups of dwarf mongooses that were used in the suggested method. The family hunts as a unit, and the main female starts hunting. This sets the sleeping mounds, the road for hunting, and the distance that is covered. The EDMO-NLS method shown here finds nodes whose locations are unknown in UWSN. The modeling results showed that the proposed model worked better than existing methods. Based on the findings and the discussion, it is clear that the suggested method has shown the best localization success in UWSN