Modify Fancy Chapter Heading to show Chapter Name

Question

In the code given below, I wanted to have the chapter names show up instead of the numbers just for the first two chapters. This code is building from the code found here.

I am writing a "Problem" for chapter 1, and the "Solution" for chapter 2. So I wanted to have "Problem" appear on the chapter header where the number 1 is for the first chapter, and then "Solution" show up where the number 2 is for the second chapter. For the List of Tables, List of Figures etc., and all the other chapters in the appendix, I wanted to leave the headers as it is currently. Can you help me modify the code so that just the first two chapters show the names in place of the numbers like this for chapter 1:

Here is the code:

\documentclass{book}
\usepackage{tikz}
\usepackage{fourier}
\usepackage[explicit,calcwidth]{titlesec}
\usepackage{lipsum}

\definecolor{mybluei}{RGB}{0,173,239}
\definecolor{myblueii}{RGB}{63,200,244}
\definecolor{myblueiii}{RGB}{199,234,253}

\renewcommand\thechapter{\arabic{chapter}}

\newcommand\chapnumfont{%
  \fontsize{380}{130}\color{myblueii}\selectfont%
}

\newcommand\chapnamefont{%
  \normalfont\color{white}\scshape\small\bfseries
}

\titleformat{\chapter}
  {\normalfont\huge\filleft}
  {}
  {0pt}
  {%
  \begin{tikzpicture}[remember picture,overlay]
  \fill[myblueiii]
    (current page.north west) rectangle ([yshift=-13cm]current page.north east);
  \node[
      fill=mybluei,
      text width=2\paperwidth,
      rounded corners=6cm,
      text depth=18cm,
      anchor=center,
      inner sep=0pt] at (current page.north east) (chaptop)
    {\thechapter};%
    {};
  \node[
      anchor=south east,
      inner sep=0pt,
      outer sep=0pt] (chapnum) at ([xshift=-20pt]chaptop.south)
    {\chapnumfont\thechapter};
    {};
  \node[
      anchor=south,
      inner sep=0pt] (chapname) at ([yshift=2pt]chapnum.south)
  {};
  \node[
      anchor=north east,
      align=right,
      inner xsep=0pt] at ([yshift=-0.5cm]chapname.east|-chapnum.south)
  {\parbox{.7\textwidth}{\raggedleft#1}};
  \end{tikzpicture}%
  }

\titleformat{name=\chapter,numberless}
  {\normalfont\huge\filleft}
  {}
  {0pt}
  {\begin{tikzpicture}[remember picture,overlay]
  \fill[myblueiii]
    (current page.north west) rectangle ([yshift=-13cm]current page.north east);
  \node[
      fill=mybluei,
      text width=2\paperwidth,
      rounded corners=6cm,
      text depth=18cm,
      anchor=center,
      inner sep=0pt] at (current page.north east) (chaptop)
    {};%
  \node[
      anchor=south east,minimum width=2in,
      inner sep=0pt,
      outer sep=0pt] (chapnum) at ([xshift=-20pt]chaptop.south)
    {};
  \node[
      anchor=south,
      inner sep=0pt] (chapname) at ([yshift=2pt]chapnum.south)
  {};
  \node[
      anchor=north east,
      align=right,
      inner xsep=0pt] at ([yshift=-0.5cm]chapname.east|-chapnum.south)
  {\parbox{.7\textwidth}{\raggedleft#1}};
  \end{tikzpicture}%
  }

\begin{document}

\chapter{Problem}

\lipsum[1]
\section{Problem Statement 1}
\lipsum[1]

\chapter{Solution}
\section{Solution Statement 1}
\lipsum[1]
\section{Solution Statement 2}
\lipsum[1]

\renewcommand{\chaptername}{Appendix}
\renewcommand{\leftmark}{\sffamily\normalsize\bfseries       
\appendixname~\thechapter}
\appendix
\chapter{References}

\end{document} 

Answer 1

The point at which the chapter text starts seems to be fixed at about 4cm below the top of the text area. I adjusted \fontsize and 1.5\titleheight to taste, so feel free to adjust them too.

Note: The curved shape is achieved by creating a box with rounded corners of which only the lower left quadrant is on the paper.

\documentclass{book}
\usepackage{tikz}
\usepackage{fourier}
\usepackage[explicit,calcwidth]{titlesec}
\usepackage{lipsum}

\definecolor{mybluei}{RGB}{0,173,239}
\definecolor{myblueii}{RGB}{63,200,244}
\definecolor{myblueiii}{RGB}{199,234,253}

\newlength{\titleheight}
\setlength{\titleheight}{\dimexpr 1in+\topmargin+\headheight+\headsep+4cm}

\titleformat{\chapter}{}{}{0pt}{%
  \begin{tikzpicture}[remember picture,overlay]
  \fill[myblueiii]
    (current page.north west) rectangle ([yshift=-\titleheight]current page.north east);
  \node[
      fill=mybluei,
      text width=2\paperwidth,
      rounded corners=6cm,
      text height=1.5\titleheight,
      anchor=center,
      inner sep=0pt] at (current page.north east) (chaptop) {};
  \node[above left, inner xsep=0pt] at (chaptop.south)
    {\fontsize{90}{90}\color{white}\scshape\bfseries #1};
  \end{tikzpicture}%
  }

\begin{document}

\chapter{Problem}

\lipsum[1]
\section{Problem Statement 1}
\lipsum[1]

\chapter{Solution}
\section{Solution Statement 1}
\lipsum[1]
\section{Solution Statement 2}
\lipsum[1]

\appendix
\chapter{Appendix}
\lipsum[1]
\chapter{References}
\lipsum[1]
\end{document}

\end{document} 

\end{document}