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In this article, an effective numerical technique for solving the variable order Fredholm–Volterra integro-differential equations (VO-FV-IDEs), systems of VO-FV-IDEs and variable order Volterra partial integro-differential equations (VO-V-PIDEs) is given. The suggested technique is built on the combination of the spectral collocation method with some types of operational matrices of the variable order fractional differentiation and integration of the shifted fractional Gegenbauer polynomials (SFGPs). The proposed technique reduces the considered problems to systems of algebraic equations that are straightforward to solve. The error bound estimation of using SFGPs is discussed. Finally, the suggested technique’s authenticity and efficacy are tested via presenting several numerical applications. Comparisons between the outcomes achieved by implementing the proposed method with other numerical methods in the existing literature are held, the obtained numerical results of these applications reveal the high precision and performance of the proposed method.

A numerical treatment to a system of Caputo fractional order differential- algebraic equations (SFDAEs) is presented throughout this article. The suggested method based upon the shifted Chebyshev pesedu- spectral method (SCPSM). The shifted Chebyshev polynomials (SCPs) are handled to reduce the SFDAEs into the solution of linear/ nonlinear systems of algebraic equations. By using some tested applications, the effectiveness and the accuracy of the suggested approach are demonstrated graphically. Also numerical comparisons between the proposed technique with other numerical methods in the existing literature are held. The numerical results show that the proposed technique is computationally efficient, accurate and easy to implement.

In this article, a novel shifted Gegenbauer operational matrix (SGOM) of fractional derivative in the Caputo sense is derived. Based on this operational matrix, an accurate and effective numerical algorithm is proposed. The SGOM of fractional derivative in conjunction with the tau method are used for solving the constant and variable coefficients space–time fractional telegraph equations (FTE) with various types of boundary conditions, namely, Neumann, Dirichlet and Robin conditions. The convergence analysis of the proposed method is established in $$\mathcal {L}^{2}_{\omega _{\alpha }}$$ . Finally, miscellaneous test examples are given and compared with other methods to clarify the accuracy and efficiency of the presented algorithm.

This paper declares a new spectral collocation technique to provide accurate approximate solutions of the one- and two-dimensional time and space fractional coupled Burgers’ equations (TSFCBEs) in which the fractional derivatives are defined according to Caputo’s definition. The suggested method is based on the shifted Gegenbauer polynomials (SGPs) for approximating the solution of TSFCBEs. The suggested technique reduces the considered problems to the solution of nonlinear algebraic equations (NLAEs). Moreover, the accuracy and reliability of the proposed method are confirmed through numerical examples. Finally, the obtained numerical results are compared with those previously reported in the literature.

In this article, we use the operation matrix (OM) of Riemann–Liouville fractional integral of the shifted Gegenbauer polynomials with the Lagrange multiplier method to provide efficient numerical solutions to the multi-dimensional fractional optimal control problems. The proposed technique transforms the under consideration problems into sets of nonlinear equations which are easy to solve. Numerical tests including numerical comparisons with some existing methods are introduced to demonstrate the accuracy and efficiency of the suggested technique.

This paper introduces a new numerical mechanism for solving multi- order fractional di erential equations (MOFDEs)and systems of fractional di erential equations, in which the fractional derivatives are expressed in Riemman- liouville(RL) sense. A new shifted ultraspherical (Gegenbauer) operational matrix (SGOM) of fractional integration of arbitraryorder is induced. By using this matrix jointly with the Tau method, the solution of fractional di erential equation (FDE)is decreased to the solution of a system of algebraic equations (AEs). Helpful problems are built-in to show the powerfuland validity of the proposed technique.

Through this article, a numerical scheme based upon the modified fractional Euler method (MFEM) is introduced to find the numericalsolutions of linear and nonlinear systems of fractional differential equations (SFDEs) as well as nonlinear multi-order fractionaldifferential equations (MOFDEs). The fractional derivatives are defined by Caputo. The proposed algorithm is very simple and providesthe solutions directly without linearization, perturbations or any other assumptions. Illustrating examples with numerical comparisonsbetween the proposed algorithm and the exact and/or fourth order Runge Kutta method (RK4) are given to reveal the efficiency and theaccuracy of our algorithm.

In this paper, by introducing the fractional derivative in sense of Caputo, the Laplace- variational iteration method (LVIM) and the Laplace-Adomian decomposition method (LADM) are directly extended to study the linear and nonlinear systems of fractional partial differential equations, as a result the approximated numerical solutions are acquired in the form of rapidly convergent series with easily computable components. Numerical results show that the two approaches are easy to implement and accurate when are applied. Compassions are made between the two methods and exact solutions. Figures are used to show the efficiency as well as the accuracy of the achieved approximated results.

This paper presents a method to solve numerically the linear quadratic optimal control problems (LQOCPs) of the fractional singular control systems with Caputo fractional derivatives. Under certain conditions on the singular fractional control systems, the problems under consideration are transformed into LQOCPs of the standard normal fractional control systems. Based on the Grunwald–Letnikov approximation (GLA) of the fractional derivatives (FDs), a numerical technique is used to solve the LQOCPs of the standard normal fractional control systems. An illustrative example is introduced to demonstrate the applicability of the proposed technique.

We solve the one- and two-dimensional fractional coupled Burgers’ equations (FCBEs) by three different methods. The proposed methods are the Laplace–Adomian decomposition method (LADM), the Laplace-variational iteration method (LVIM) and the reduced differential transform method (RDTM). The solutions are obtained as rapidly convergent series with simply calculable terms. Numerical studies of the application of these approaches for a number of sample problems are given and are illustrated graphically. With these methods, it is possible to investigate the nature of solutions when the fractional derivative parameters are changed. The numerical results reveal the effectiveness and the correctness of the proposed methods.

The performance of the new EGM2008 global geopotential model (GGM), over the northern Nile valley and delta in Egypt, has been evaluated. Over 305 global positioning system (GPS)/levelling stations, the standard deviation of the undulation differences has been estimated to be 0.23 m. Thus, it can be concluded that the EGM2008 is superior to earlier GGMs over the northern area of Egypt by a factor of 1.6. Moreover, four mathematical models have been investigated in order to incorporate local data sets with the EGM2008, leading to the conclusion that the first-order polynomial is the optimum one. An improvement of about 26% has been obtained after applying that regression model, and the standard deviation has been decreased to 0.17 m. As a result, the EGM2008 produces a crucial influence on integrated GPS surveys where the orthometric heights can be obtained without any additional costs. Such a practice presents a suitable alternative, from an economical point of view, to substitute the expensive traditiona...

مع انتشار تطبيقات النظام العالمي لتحديد المواقع في مشروعات المساحة والخرائط عالميا، فقد تزايدت الحاجة في مصر للحصول على نموذج دقيق لسطح الجيويد، وخاصة لتحويل الارتفاعات الجيوديسية إلي المناسيب المناظرة. يتم استنباط نماذج "شبيه الجيويد" تقليديا باستخدام قياسات الجاذبية الأرضية، ثم يتم ضبطها لتتوافق مع نقاط معلوم عندها قيم حيود الجيويد الناتجة من النظام العالمي لتحديد المواقع والميزانيات الأرضية. لكن يجب قبل إجراء عملية التوافق أن يتم تحويل نموذج شبيه الجيويد إلى نموذج جيويد حقيقي، وهذه الخطوة لم تؤخذ في الاعتبار في معظم الدراسات الجيوديسية في مصر. وبناء على قياسات جيوديسية حقيقية عند 1080 نقطة فقد تم حساب المسافات بين شبيه الجيويد و الجيويد ووجد أن قيمها تتراوح بين -0.074 متر و 0.367 متر بمتوسط يبلغ 0.060 متر. وبعد إجراء هذا التحويل فان الدقة المتوسطة لأحد نماذج الجيويد العالمية - التي تم اختبارها ووجد أنه أحسن نموذج عالمي لتمثيل مجال الجاذبية في مصر - قد تحسنت من 0.52 متر إلى 0.46 متر بتحسن يبلغ 11 ? في المتوسط. وتدل النتائج التي تم الحصول عليها على أن عملية التحويل من نماذج شبيه الجيويد إلي نماذج الجيويد تعد خطوة مؤثرة ويجب أخذها في الاعتبار في أية محاولة مستقبلية لتطوير نموذج جيويد دقيق في مصر.