| Summary: | Integral equations play an important role for their applications in practical engineering and applied science, and nonlinear Urysohn integral equations can be applied when solving many problems in physics, potential theory and electrostatics, engineering, and economics. The aim of this paper is the use of spline quasi-interpolating operators in the space of splines of degree <i>d</i> and of class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>C</mi><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></semantics></math></inline-formula> on uniform partitions of a bounded interval for the numerical solution of Urysohn integral equations, by using a superconvergent Nyström method. Firstly, we generate the approximate solution and we obtain outcomes pertaining to the convergence orders. Additionally, we examine the iterative version of the method. In particular, we prove that the convergence order is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mi>d</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> if <i>d</i> is odd and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mi>d</mi><mo>+</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula> if <i>d</i> is even. In case of even degrees, we show that the convergence order of the iterated solution increases to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mi>d</mi><mo>+</mo><mn>4</mn><mo>)</mo></mrow></semantics></math></inline-formula>. Finally, we conduct numerical tests that validate the theoretical findings.
|
|---|