Semi-Analytical Methods for Differential Equations

Venue: Yarmouk University, Jordan

Dates: April 8-17, 2025.

Inscription Deadline: 24 March 2025.

Coordinators

Abdeljalil Nachaoui, Nantes Université (Abdeljalil.Nachaoui@univ-nantes.fr) & Fatima M. Aboud, University of Diyala, Iraq (Fatima.Aboud@uodiyala.edu.iq).

Sponsors

Scientific committee 

  • Mr. Abdeljalil Nachaoui, Laboratoire de Mathématiques Jean Leray, Nantes Université, France, (Abdeljalil.Nachaoui@univ-nantes.fr)
  • Mr. Tamaz Tadumdaze, Institute of Applied Mathematics, Tbilisi State University, Tbilisi,Georgia, (tamaz.tadumadze@tsu.ge)
  • Mr. Francois Jauberteau, Laboratoire de Mathématiques Jean Leray, Nantes Université, France, (Francois.Jauberteau@univ-nantes.fr)
  • Mrs. Sharifa Alsharif, Department of Mathematics, Faculty of Science, Yarmouk University, Jordan,sharifa@yu.edu.jo.
  • Mrs. Gulnar Sadiq, Department of Mathematics, College of Education, University of Sulaimani, Sulaymaniyah, Iraq. (gulnar.sadiq@univsul.edu.iq)
  • Mr. Mohammad Alrifai, Department of Mathematics, Faculty of Science, Yarmouk University, Irbid, Jordan, m_alrefai.edu.jo
  • Mr. Edris Rawashdeh, Department of Mathematics, Faculty of Science, Yarmouk University, Irbid, Jordan, edris@yu.edu.jo

 Organizing committee

  • The person in charge: Mrs. F. Aboud, Department of Mathematics, College of Sciences, University of Diyala, Iraq (Fatima.Aboud@uodiyala.edu.iq)
  • Mr. Ahmad Baniabedalruhman, Department of Mathematics, Faculty of Science, Yarmouk University, Irbid, Jordan (ahmad_a@yu.edu.jo).
  • Mr. Ghassan E. Arif, Tikrit University, Tikrit, Iraq , (ghasanarif@tu.edu.iq).
  • Dr. Basem Alkhamaiseh, Yarmouk University, Irbid, Jordan, (Mathematics. basem.m@yu.edu.jo).
  • Mrs. Sharifa Alsharif, Yarmouk University, Irbid , Jordan, (sharifa@yu.edu.jo).
  • Mr. Rashid Abu-Dawas, Yarmouk University, Irbid, Jordan (rrashid@yu.edu.jo).
  • Mr. Mohammad F. Al-Jamal, Yarmouk University, Irbid, Jordan (mfaljamal@yu.edu.jo).
  • Mrs. Gulnar Sadiq, Department of Mathematics, College of Education, University of Sulaimani, Sulaymaniyah, Iraq. (gulnar.sadiq@univsul.edu.iq)
  • Mrs. Wafaa Fayeq Ghaidan, Department of Mathematics, College of Sciences, University of Diyala, Iraq (wafaa_fayeq@uodiyala.edu.iq)
  • Mr. Mohammad Shakhatreh, Yarmouk University, Irbid, Jordan (mali@yu.edu.jo).
  • Mr. Nedal AlAnaghreh, Irbid private University, Irbid, Jordan (alanaghreh_nedal@yahoo.com).
  • Mr. Samer Alokaily, University of Petra, Amman, Jordan (samer.alokaily@uop.edu.jo)

 Scientific contents

Lecturers and courses titles

  Description of each course

 Course of Fatima ABOUD: Weighted Residual Methods. 

The weighted residual method (WRM) is a powerful technique for computing solutions to differential equations with boundary conditions, known as boundary value problems (BVPs). WRM is an approximation method where the solution of a differential equation is represented as a linear combination of trial or shape functions with unknown coefficients. The approximate solution is substituted into the governing differential equation, resulting in an error or residual. In WRM, this residual is forced to vanish at average points or minimized based on the weight function to determine the unknown coefficients. 

Course of  Mohammed Al-Refai: Recent Advances in Fractional Operators and Associated Fractional Differential Equations

This course explore different types of fractional operators with both singular and nonsingular kernels. Emphasis is placed on recent progress in fractional calculus, including normalized fractional operators, those with nonsingular kernels, and general fractional operators. We then discuss the solvability of related fractional Cauchy problems with various fractional differential operators.  Since fractional derivatives with nonsingular kernels vanish at the origin, additional conditions were imposed in the corresponding fractional differential equations to ensure the existence of solutions. To tackle this challenge, we introduce an extension of the fractional operator to wider function spaces, permitting an integrable singular kernel at the origin. Additionally, we explore various methods to overcome the limitations of fractional operators with nonsingular kernels. We will explore the concept of general fractional operators and outline potential directions for future research.

 

Course of A. K Alomari: Fractional differential equations and fractional calculus. 

This course provides a comprehensive introduction to fractional differential equations and fractional calculus, offering participants a deep dive into these advanced mathematical concepts and their applications. Participants will begin with a foundational overview of fractional calculus, exploring key definitions, properties, and theorems. The tutorial will cover various types of fractional derivatives and integrals, such as the Riemann-Liouville, Caputo, and Grunwald Letnikov formulations. Attendees will also learn about methods for solving both linear and nonlinear fractional differential equations and numerical approaches will be covered in detail. 

Course of Abdeljalil Nachaoui : Wavelet-Collocation Approximation for Differential and Partial Differential Equations 

Haar wavelets are a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. They are used in various applications, including solving differential and partial differential equations (PDEs). 

This tutorial provides an introduction to Haar wavelets and their application in approximating solutions to differential and partial differential equations. The implementation of the discrete problems will be done using MATLAB. 

Course of Sudad Rasheed: Polynomial Expansion approximation. 

The course will cover the application of polynomial expansions to solve ODEs and PDEs. We will introduce the basic principles of polynomial expansions, including a review of key concepts and properties. Participants will learn about the construction and use of different types of polynomial series, such as Taylor series and Chebyshev polynomials, focusing on their convergence properties and how they can be applied to approximate functions. They will also have the opportunity to work through problem sets to solidify their understanding and gain hands-on experience. 

Practical part of courses: Professor Nachaoui will supervise, in addition to his course, the part of implementation of numerical methods for other courses 

   Tentative schedule 

To cover these courses, we will need 24 hours for theoretical part and 24 hours for practical part. In the end of the course there will be theoretical and practical evaluations. The doctoral students will have the possibility to speak about their interesting domain of research, there will be 6 hours of communications 

  General Introduction. 

  • Motivation, examples from physical systems in various scientific and engineering disciplines 

  Fractional differential equations and fractional calculus 

  • Practical applications of fractional calculus in diverse fields
  • Overview of fractional calculus
  • Methods for solving both linear and nonlinear fractional differential equations.
  • Stability analysis, and the interpretation of fractional models 

    Weighted Residual Methods 

  • Introduction
  • Collocation Method
  • Subdomain Method
  • Least-square Method
  • Comparison of WRMs 

    Polynomial Expansion approximation. 

  • Introduction to the basic principles of polynomial expansions
  • Taylor series, Fourier series, and Chebyshev polynomials
  • How polynomial expansions can simplify and solve complex differential equations
  • Numerical implementation 

    Wavelet-Collocation Approximation for Differential and Partial Differential Equations 

  • Introduction
  • Discretize the Equation: Transform the differential equation into a discrete form suitable for wavelet analysis.
  • Apply Wavelet Transform: Express the functions involved in the equation using Haar wavelets.
  • Formulate the Problem: Convert the original problem into a system of algebraic equations.
  • Solve the System: Use numerical methods to solve the resulting system for the wavelet coefficients. 

   Programming methods with MATLAB.