# Department of Applied Mathematics

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### Recent Submissions

Now showing 1 - 5 of 5
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Solving Jeffery–Hamel Problem with Variable Magnetic Field and its Instability Using the Modified Adomian Decomposition Method
(University of Khartoum, 2021) Salsabeel Salah Mohamed Hamdan
Abstract The problem of the magneto-fluid-dynamics boundary layer on Internal flow between two plates is important in many areas, and many methods have been used to solve it. The governing equation to be solve is ( ) ( ) ( ) ( ) ( ) , The objective of this research is to apply the new Modified Adomian Decomposition Method (MADM) to solve the problem. The MADM is uses a standard method of decomposition. We have found that it is more advantageous to all previous methods. It provided a more accurate method for the drag.
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An Improved Differential Transformation Method for Solving Initial Value Problems
(University of Khartoum, 2021) Mohammed Abdelhalim Said Mahmood
ABSTRACT Most of the problems in mathematics and nature are inherently nonlinear. The Differential Transformation Method (DTM), is one of the most effective semi-numericalanalytical methods for solving and analyzing these problems, Moreover, it can solve all types of Differential Equations. The main problems in the (DTM) is the need for higher orders to get an accurate solution, and the resulting polynomial solution always has a large truncation error. The statement of this thesis is highlighted on finding a way to reduce the order of the (DTM) while maintaining the same error limits. In this thesis an improved (DTM) is designed -aimed at solving (ODE-IVPs)- to reduce the order of (DTM) while maintaining the same accuracy. In order to fulfill this aim two approaches are used to attain the improvement. The first approach is based on using Telescoping Procedures for Power Series (TPPS) method, the resulting order reduction method is called Telescoping Order Differential Transformation Method (TO-DTM),here, the (TPPS) method has been generalized, improved, appropriated, and applied in the (DTM) to reduce its order. In order to do this, two theorems have been prepared and proved, and appropriate error analysis is carried out. The second approach relieson the Least Squares Polynomial Method (LSPM), and the resulting order reduction method is called Least Squares Differential Transformation Method (LS-DTM), here, the (LSPM) has been improved, appropriated, and applied in the (DTM) to reduce its order. The (TO-DTM), and the (LS-DTM) are compared against the original (DTM) and the recently known method called the Residual Power Series Method (RPSM). The comparison is directed to answering the question about whether the (TO-DTM) and (LS-DTM) methods will be better than the (DTM) and (RPSM), in terms of accuracy, easiness, and effciency. Five test problems of different types of (IVPs), i.e. (first, second, third order, linear, nonlinear, system, and differential algebraic equations (DAEs)) are used. The major result obtained is that our two approaches had reduced the order from 45 to 10 while (TO-DTM) achieving a better accuracy and effciency, followed by the (LS-DTM). As for the (DTM) and (RPSM), both of them were the same in the criterion of e ciency and accuracy. As for the term of easiness, the (DTM) it was the better, followed by (RPSM), then (LS-DTM), then (TO-DTM). المستخلص معظم المسائل في الرياضيات والطبيعة غير خطية بطبيعتها، تعد طريقة التحويلات التفاضلية واحدة من أكثر الطرق شبه-العددية-التحليلية فاعلية في حل هذه المسائل، أضف إلى (DTM) هي (DTM) ذلكأنه يمكن استخدامها لحل جميع أنواع المعادلات التفاضلية. المشكلة الرئيسية في الحاجة لرتب عالية منها للحصول على حل دقيق، وكثيرة الحدود الناتجة كحل تقريبي دائماً تكون ذات خطأ اقتطاع كبير. تم تسليط الضوء في هذه الأطروحة على إيجاد طريقة لتخفيض مع الحفاظ على نفس حدود الخطأ. حيث تم في هذه الأطروحة تصميم تحسين (DTM) رتبة (IVPs) لتقليل رتبتها مع الحفاظعلى نفسالدقة، بهدفحل مسائل القيم الابتدائية (DTM) لطريقة لتحقيق هذا الهدف تم استخدام طريقتين لعمل التحسين. .(ODEs) في المعادلات التقاضلية العادية سميت الطريقة المخفضة ،(TPPS) اعتمدت الطريقة الأولى على طريقة التلسكوب لسلاسل القوى ،(TPPS) حيث تم هنا تعميم طريقة ،(TO-DTM) الناتجة بطريقة التحويل التفاضلي مخفضة الرتبة لتخفيض رتبتها. ومن اجل ذلك تم (DTM) ومن ثم تحسينها وتخصيصها وتطبيقها في طريقة تقديم نظريتين باثباتاتهما، كما تم إجراء تحليل مناسب للخطأ. الطريقة الثانية اعتمدت على سميت الطريقة المخفضة الناتجة بطريقة التحويل التفاضلي .(LSPM) طريقة المربعات الصغرى ومن ثم تخصيصها وتطبيقها في ،(LSPM) حيث تم تحسين طريقة ،(LS-DTM) بالمربعات الصغرى (LS-DTM) وطريقة ،(TO-DTM) لتخفيضرتبتها.كما تمت مقارنةكل من طريقة (DTM) طريقة الاصلية و الطريقة المعروفة مؤخراً والتي تسمى طريقة متبقي سلاسل القوى (DTM) في مقابل (LS-DTM) و (TO-DTM) المقارنة موجهة للإجابة على التساؤل حول ما إذاكانت طريقتا ،(RPSM) من حيث الدقة والسهولة والكفاءة. تم استخدام خمس (RPSM) و (DTM) ستكونان أفضل من اعني معادلات تفاضلية من الرتب الأولى والثانية ،(IVPs) مسائل اختبارية من أنواع مختلفة من والثالثة، ومعادلات تفاضلية خطية وغير خطية، وانظمة معادلات تفاضلية، ومعادلات تفاضلية (LS-DTM) و (TO-DTM) جبرية. النتيجة الرئيسية التي تم الحصول عليها هي أن كل من طريقتي هي الأفضل من حيث الكفاءة والدقة، (TO-DTM) قد قللت الرتبة من 45 إلى 10 ، حيث كانت طريقة فقدكانتا متماثلتين في معيار (RPSM) و (DTM) أما بالنسبة للطريقتين ،(LS-DTM) تليها طريقة هي الأسهل متبوعة (DTM) الكفاءة والدقة. اما فيما يتعلق بمعيار السهولة فقد كانت طريقة . (TO-DTM) ثم (LS-DTM) ثم تلتهما طريقة (RPSM) بطريقة
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A Mathematical Analysis for the Dynamics of Meningitis Disease with and without Co-infection
(University of Khartoum, 2021-04) Salma Omar Abdalmageid Adam
Abstract Meningitis is inﬂammation of the meninges, the covering of the brain and spinal cord. It is most often caused by infection (bacterial, viral, or fungal), but can also be produced by chemical irritation, cancer and other conditions. Bacterial meningitis is very serious and can be deadly. The fatality rate is very high in Africa Meningitis Belt. It is approximately 20%. Up to 20% of the survivors get serious disabilities as a result of the disease. Current models ignore the existence of recovered individuals with disabilities. In this thesis a mathematical model of transmission of meningitis disease is built to study the impact of vaccination in the presence of survivors with disabilities. Another model is made to study the effect of co-infection with TB disease. Deterministic mathematical models are created. In these models the whole population is divided into six classes; susceptible, vaccinated, carriers, infected, recovered without disabilities and recovered with disabilities. The dy- namical behavior is studied with ﬁxed control for both vaccine and treatment. Then these rates are used as control functions in order to obtain a strategy that minimizes the num- ber of infected while keeping the cost of vaccination and treatment as low as possible. Pontryagins Maximum Principle method is applied in that respect. Also a deterministic mathematical model was created to study a co-infection between meningitis disease and TB disease. The total population is divided into eleven compartments representing the interaction between the two diseases. The two sub-models are studied, separately, ﬁrst. Then the full joint model is considered. The analysis of these models showed that the disease-free equilibrium is globally asymptotically stable. Also the endemic equilibrium exists and is unique but under some conditions. Also if the rate of vaccine uptake and the vaccine efﬁcacy rate are kept at some values then the disease can be controlled and the disabilities of survivors could be reduced. In the presence of optimal control measures, it is shown that in approximately 20 years, the number of carriers will decrease and also the number of recovered individuals with disabilities will vanish. Moreover, the number of infected cases will noticeably decline and ﬁnally vanish in approximately 12.5 years. In the absence of the control measures the number of individuals in all classes is going to increase and the infection in the absence of the prevention will be hard to control. For the meningitis sub-model it was found that the disease free equilibrium is locally asymptoti- cally stable if the basic reproductive number R0M < 1 and also it is globally asymptotically stable. The endemic equilibrium exists but is locally unstable. In the TB sub-model, it is found that the disease free equilibrium is locally asymptotically stable if R0T < 1 and also it is globally asymptotically stable. The endemic equilibrium exists and is locally asymp- totically stable. The full model have shown that R0 = max{R0M, R0T}, and the disease free equilibrium E0 is locally asymptotically stable if R0T < 1. Our numerical simulation have shown that when the basic reproductive numbers of both diseases are less than unity then the infections could be controlled and die out of the community. But if one of the basic reproductive numbers or both are greater than unity then it is found that one or both diseases are not controlled.
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Development of Spectral and Pseudospectral Methods for Solving Partial Differential Equations with time-delays
(University of Khartoum, 2016-03-22) Ahmed Mohamed Abdalla Adam ; Mohsin H. A. Hashim ; Department of Applied Mathematics
This thesis is concerned with the development of numerical methods for solving delay partial differential equations. Delay partial differential equations do arise in various applications, like biology for instance, -in population dynamics and epidemiology-, medicine, control theory, climate models, and many others. Due to the complex structure of their governing equations, analytical solutions are not available in general, and therefore one has to rely mostly on some numerical methods. The classical numerical approaches are inefficient for this type of problems because of their unrealistic stability conditions. In this thesis four spectral methods to solve different classes of delay problems are developed. These problems include single equation or system of equations. The first of these methods is a fitted Fourier pseudospectral method for solving two delayed diffusive population models. The second is a fitted Chebyshev pseudospectral method for delayed diffusive model. The third is a Fitted Galerkin's spectral method for delayed diffusive and delayed reaction-diffusion models. The fourth one is Chebyshev pseudospectral method for solving a system of delay differential equations describing co-operation reaction diffusion population model. Results show that these numerical methods are highly stable with high order of convergence. Comparison of these methods to those available in literature, reﬂects that these methods are reliable, accurate and efficient. هذا البحث يتعلق بتطوير طرق عددية لحل المعادلات التفاضلية الجزئية المتأخرة. المعادلات التفاضلية الجزئية المتأخرة تنتج (تنشأ) من تطبيقات متنوعه، مثل علم الأحياء على سبيل المثال- في ديناميكا السكان و علم الوبائيات- الطب، نظرية التحكم، نماذج المناخ و غيرها الكثير. نسبة للتركيبة المعقده للمعادلات الحاكمة لها، الحلول التحليلية ليست متاحه على العموم، و لذلك غالباً ما نعول على بعض الحلول العددية. الطرق العددية المتعارف عليها عديمة الكفاءة لهذه الأنواع من المسائل لعدم واقعيه شروط استقرارها. في هذا البحث طورت أربع طرق طيفيه لحل أصناف مختلفة من مسائل التأخر. هذه المسائل تتضمن معادلة وحيدة أو منظومة معادلات. أولي هذه الطرق هي طريقة فوريير الطيفيه الزائفه الملائمه لحل اثنين من نماذج الانتشار السكاني المتأخر. ثانيها طريقة شبشيف الطيفيه الزائفه الملائمه لحل نموذج الانتشار المتأخر. ثالثها طريقة جليركن الطيفيه الملائمه لحل نموذجي الانتشار و الانتشار-التفاعل المتأخرين. رابعها طريقة شبشيف الطيفيه الزائفه لحل منظمومة معادلات تفاضلية متأخرة تصف نموذج الانتشار-التفاعل لتعاون السكان. النتائج أظهرت أن هذه الطرق العددية عالية الاستقرار مع علو رتبة التقارب. مقارنة هذه الطرق مع تلك التي أنجزت في الدراسات السابقة، عكست أن هذه الطرق يعول عليها، دقيقه و ذو كفاءة عالية
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A Derivative-Free Algorithm for Linearly Constrained Optimization Problems
(University of Khartoum, 2015-04-20) Elzain Ahmed Elzain Gumma ; Mohsin Hassan Hashim ; Faculty of Mathematical Sciences
Derivative-free optimization is an active area of research, because there are many practical problems for which the derivatives are not available, and it may still be desirable to carry out optimization. The main motivation for the study of such problems is the high demand for the solution for such problems. In this thesis a new derivative-free algorithm has been developed, named LCOBYQA. The main aim of this algorithm is to nd a minimum x? 2 Rn of a nonlinear objective subject to linearly inequality constraints. The algorithm is based on the trust region method, and uses well known techniques such as the active set version of truncated conjugate gradient method, multivariate Lagrange polynomial interpolation, and QR factorization. Each iteration of the algorithm constructs a quadratic approximation (model) of the objective function that satis es interpolation conditions and leaves some freedom in the model, taken up by minimizing the Frobenius norm of the change of the second derivative of the model. A typical iteration of the algorithm generates a new vector of variables either by minimizing the quadratic model subject to the given constraints and the trust region bound, or by a procedure that should improve the accuracy of the model. Numerical results show that LCOBYQA works well and is so competing against available model-based derivative-free algorithms, such as CONDOR, COBYLA, UOBYQA, NEWUOA and DFO. Under certain conditions LCOBYQA is observed to work extremmely and amazingly fast, leaving an open further investigation to be considered.