• Basic information
• Learning outcomes
• Contents
• Assessment
• Workload

Learning outcomes

Upon successful completion of this course, students will be able to:

1. Explain the Fundamental concepts of the Theory of Integral Equation.

   Contribution to Program Outcomes

   1. Having the knowledge about the scope, applications, history, problems, and methodology of mathematics that are useful to humanity both as a scientific and as an intellectual discipline.
   2. Having improved abilities in mathematics communications, problem-solving, and brainstorming skills.

   Method of assessment

   1. Written exam
2. Distinguish the difference between Differential Equations and Integral Equations.

   Contribution to Program Outcomes

   1. Having the knowledge about the scope, applications, history, problems, and methodology of mathematics that are useful to humanity both as a scientific and as an intellectual discipline.
   2. Describing, formulating, and analyzing real-life problems using mathematical and statistical techniques.
   3. Having improved abilities in mathematics communications, problem-solving, and brainstorming skills.

   Method of assessment

   1. Written exam
   2. Homework assignment
3. Develop awareness for the Integral Equations.

   Contribution to Program Outcomes

   1. Having the knowledge about the scope, applications, history, problems, and methodology of mathematics that are useful to humanity both as a scientific and as an intellectual discipline.
   2. Having improved abilities in mathematics communications, problem-solving, and brainstorming skills.
   3. Exhibiting professional and ethical responsibility.

   Method of assessment

   1. Written exam
   2. Homework assignment

Contents
Week 1:	Fredholm equations. Concept of integral equations.
Week 2:	Fredholm operator and its degree. Iterated kernel. Method of successive approximations.
Week 3:	Volterra equation. Concept of resolvent. Integral equations with degenerated kernels.
Week 4:	General case of Fredholm equation. Conjugate Fredholm equation. Fredholm theorems. Resolvent. The case of several independent variables. Equations with weak singularity. Continuous solutions of integral equations. Systems of integral equations. Examples of non-Fredholm integral equations.Riesz-Schauder equations. Fundamental concepts of operators. Method of successive approximations for equations with conjugate bounded operators. Completely continuous operators. Solution of Riesz-Schauder equations. Extension of Fredholm theorems. Symmetric integral equations. Symmetric kernels. Fundamental theorems on symmetric equations. Theorem on existence of a characteristic constant. Hilbert-Schmidt theorem. Solution of symmetric integral equations. Bilinear series. Bilinear series for iterated kernel. Resolvent of a symmetric kernel.Extremal properties of characteristic constants and proper functions. Applications of integral equations. Integral equations of potential theory in the three dimensional space. Solution of boundary value problems of the potential theory. Solution of the Dirichlet exterior problem. Equations of the plane potential theory. Boundary value problem for an ordinary differential equation. Characteristic constants and proper functions of an ordinary differential operator. Proof the Fourier method. Green function for the Laplace operator. Proper functions of the problem on vibrations of a membrane.
Week 5:	The case of several independent variables. Equations with weak singularity. Continuous solutions of integral equations.
Week 6:	MIDTERM EXAM I and Solutions
Week 7:	Systems of integral equations. Examples of non-Fredholm integral equations.
Week 8:	Riesz-Schauder equations. Fundamental concepts of operators. Method of successive approximations for equations with conjugate bounded operators. Completely continuous operators. Solution of Riesz-Schauder equations. Extension of Fredholm theorems. Symmetric integral equations. Symmetric kernels. Fundamental theorems on symmetric equations. Theorem on existence of a characteristic constant. Hilbert-Schmidt theorem. Solution of symmetric integral equations. Bilinear series. Bilinear series for iterated kernel. Resolvent of a symmetric kernel. Extremal properties of characteristic constants and proper functions. Applications of integral equations. Integral equations of potential theory in the three dimensional space. Solution of boundary value problems of the potential theory. Solution of the Dirichlet exterior problem. Equations of the plane potential theory. Boundary value problem for an ordinary differential equation. Characteristic constants and proper functions of an ordinary differential operator. Proof the Fourier method. Green function for the Laplace operator. Proper functions of the problem on vibrations of a membrane.
Week 9:	Method of successive approximations for equations with conjugate bounded operators. Completely continuous operators.
Week 10:	Solution of Riesz-Schauder equations. Extension of Fredholm theorems. Symmetric integral equations. Symmetric kernels. Fundamental theorems on symmetric equations.
Week 11:	Theorem on existence of a characteristic constant. Hilbert-Schmidt theorem. Solution of symmetric integral equations.
Week 12:	MIDTERM EXAM II and Solutions
Week 13:	Bilinear series. Bilinear series for iterated kernel. Resolvent of a symmetric kernel. Extremal properties of characteristic Ccnstants and proper functions. Applications of integral equations. Integral equations of potential theory in the three dimensional space. Solution of boundary value problems of the potential theory. Solution of the Dirichlet exterior problem. Equations of the plane potential theory.
Week 14:	Boundary value problem for an ordinary differential equation. Characteristic constants and proper functions of an ordinary differential operator. Proof the Fourier method. Green function for the Laplace operator. Proper functions of the problem on vibrations of a membrane.
Week 15*:	-
Week 16*:	Final exam
Textbooks and materials:	Integral Equations by Abdul J. Jerri.
Recommended readings:	Linear Integral Equations by S. G. Mikhlin. Integral Equations by I.G. Petrovskii. Integral Equations by F. G. Tricomi. Integral Equations by M. Krasnov, A. Kiselev, G. Makeronko. Linear Integral Equations by Rainer Kresss. Lectures on Differential and Integral Equation by Kosaku Yosida.
	* Between 15th and 16th weeks is there a free week for students to prepare for final exam.

Assessment
Method of assessment	Week number	Weight (%)
Mid-terms:	6, 12	40
Other in-term studies:		0
Project:		0
Homework:		0
Quiz:		0
Final exam:	16	60
	Total weight:	(%)

Workload
Activity	Duration (Hours per week)	Total number of weeks	Total hours in term
Courses (Face-to-face teaching):	3	14
Own studies outside class:	3	14
Practice, Recitation:	1	14
Homework:	0	0
Term project:	0	0
Term project presentation:	0	0
Quiz:	0	0
Own study for mid-term exam:	6	2
Mid-term:	6	2
Personal studies for final exam:	6	1
Final exam:	3	1
		Total workload:
		Total ECTS credits:	*
	* ECTS credit is calculated by dividing total workload by 25. (1 ECTS = 25 work hours)