Numerical computational approach for 6th order boundary value problems

This study introduces numerical computational methods that employ fourth-kind Chebyshev polynomials as basis functions to solve sixth-order boundary value problems. The approach transforms the BVPs into a system of linear algebraic equations, expressed as unknown Chebyshev coefficients, which are subsequently solved through matrix inversion. Numerical experiments were conducted to validate the accuracy and efficiency of the technique, demonstrating its simplicity and superiority over existing solutions. The graphical representation of the method's solution is also presented

Uniform reconstruction of continuous functions with the RAFU method

[EN] The RAFU (radical functions) method can be used to obtain the uniformreconstruction of a continuous function from its values at some ofthe points of partitions of a closed interval. In this work we willprove that we can reconstruct a continuous function from average samplesof these points, from linear combinations of them and from local averagesamples given by convolution. Uniform error bounds will be established. If these data are unknown but approximate values of them are known, uniform reconstruction will be also possible. Error estimates in these cases will be given. The case of a non-uniform net will be treated. Examples and algorithms will be also shown.Corbacho Cortés, E. (2017). Uniform reconstruction of continuous functions with the RAFU method. Applied General Topology. 18(2):361-375. doi:10.4995/agt.2017.7263SWORD36137518

The Numerical Investigations of Non-Polynomial Spline for Solving Fractional Differential Equations

We present a crossing approach based on the new construction of non-polynomial spline function to investigate the numerical solution of the fractional differential equations. We find the accuracy of the spline method and to presenting the completion of non-polynomial spline two examples for problems are used. To clarify, we present the numerical computations that can be used to solve difficult problems while the results are found and got to be in good error estimation with comparing exact solutions

A Quintic B-Spline Technique for a System of Lane-Emden Equations Arising in Theoretical Physical Applications

In the present study, we introduce a collocation approach utilizing quintic B-spline functions as bases for solving systems of Lane Emden equations which have various applications in theoretical physics and astrophysics. The method derives a solution for the provided system by converting it into a set of algebraic equations with unknown coefficients, which can be easily solved to determine these coefficients. Examining the convergence theory of the proposed method reveals that it yields a fourth-order convergent approximation. It is confirmed that the outcomes are consistent with the theoretical investigation. Tables and graphs illustrate the proficiency and consistency of the proposed method. Findings validate that the newly employed method is more accurate and effective than other approaches found in the literature. All calculations have been performed using Mathematica software

AN ENHANCED WAVELET BASED METHOD FOR NUMERICAL SOLUTION OF HIGH ORDER BOUNDARY VALUE PROBLEMS

The Legendre wavelet collocation method (LWCM) is suggested in this study for solving high-order boundary value problems numerically. Eighth, tenth, and twelfth-order examples are used as test problems to ensure that the technique is efficient and accurate. In comparison to other approaches, the numerical results obtained using LWCM demonstrate that the method's accuracy is very good. The results indicate that the method requires less computational effort to achieve better results

B-Splines Based Finite Difference Schemes For Fractional Partial Differential Equations

Fractional partial differential equations (FPDEs) are considered to be the extended formulation of classical partial differential equations (PDEs). Several physical models in certain fields of sciences and engineering are more appropriately formulated in the form of FPDEs. FPDEs in general, do not have exact analytical solutions. Thus, the need to develop new numerical methods for the solutions of space and time FPDEs. This research focuses on the development of new numerical methods. Two methods based on B-splines are developed to solve linear and non-linear FPDEs. The methods are extended cubic B-spline approximation (ExCuBS) and new extended cubic B-spline approximation (NExCuBS). Both methods have the same basis functions but for the NExCuBS, a new approximation is used for the second order space derivative

طرائق شرائحية مجمعة من المرتبة الخامسة لحل معادلات تفاضلية خطية من المرتبة الثانية بشروط حدية

This paper presents numerical methods for the solution of linear second-order boundary value problems. These methods are based on C2-quintic splines, that is, fifth Hermite interpolating polynomials with three collocation points. The error analysis and sufficient conditions of the convergence for the presented methods when applied to BVPs are considered. A study shows that the proposed methods consist of order five for (c1=1/2, c2=3/4). Moreover, if: , where , then the regions of absolute stability of the methods contain some neighborhood of infinity. They are also A-stable and possess unbounded regions of absolute stability. Four widely applied problems are solved to illustrate the order and stability of the proposed methods. The comparisons of the presented methods with other methods show that our results are more accurate. يقدم هذا البحث طرائق عددية لحل مسائل القيم الحدية في المعادلات التفاضلية الخطية من المرتبة الثانية. إن الطرائق المقترحة تعتمد على كثيرات حدود هرمية الشرائحية من الدرجة الخامسة في الفضاء C2 و تحقق شروط المسألة في ثلاث نقاط مجمعة. حيث يتم تحليل الخطأ لهذه الطرائق بالإضافة إلى وضع الشروط الكافية لتقاربها لدى تطبيقها على مسائل القيم الحدية. تبين الدراسة أن الطرائق المذكورة تكون متجانسة من المرتبة الخامسة لأجل (c1=1/2, c2=3/4)، كما يشير تحليل الاستقرار إلى أنها تكون في حالة -A استقراراً وأن مناطق الاستقرار المطلق تشغل مساحات لانهائية في المستوي العقدي إذا تحققت المتراجعة: , علما بأن . وقد تم اختبار الطرائق المقترحة باستخدامها لحل أربع مسائل مطبقة على نطاق واسع، وكانت النتائج التي تم التوصل إليها دقيقة بالمقارنة مع طرائق أخرى

Numerical singular perturbation approaches based on spline approximation methods for solving problems in computational finance

Philosophiae Doctor - PhDOptions are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Bean. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature

Numerical singular perturbation approaches based on spline approximation methods for solving problems in computational finance

Philosophiae Doctor - PhDOptions are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Econ. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature.South Afric